diff b1f9bc4..56bc728 goe_rrg_bulk_universality_v2.txt
diff --git a/goe_rrg_bulk_universality_v2.txt b/goe_rrg_bulk_universality_v2.txt
index ccf2a10..9fc175e 100644
--- a/goe_rrg_bulk_universality_v2.txt
+++ b/goe_rrg_bulk_universality_v2.txt
@@ -37,21 +37,21 @@
-Fix an integer $dge 3$. Let $G_{n,d}$ be the uniform random graph
-among all simple labelled $d$-regular graphs on the vertex set
-${1,...,n}$: no loops, no repeated edges, every vertex of degree
-$d$, all such graphs equally likely. An integer $n$ is admissible
-when this set is nonempty, that is, when $nd$ is even and $dle n-1$.
-Write $A_{n,d}$ for its adjacency matrix: $A_{ij}=1$ if $i$ and $j$ are
-joined, $A_{ij}=0$ otherwise. Each row has exactly $d$ ones, so $A_{n,d}$
+Fix an integer dge 3. Let G_{n,d} be the uniform random graph
+among all simple labelled d-regular graphs on the vertex set
+{1,...,n}: no loops, no repeated edges, every vertex of degree
+d, all such graphs equally likely. An integer n is admissible
+when this set is nonempty, that is, when nd is even and dle n-1.
+Write A_{n,d} for its adjacency matrix: A_{ij}=1 if i and j are
+joined, A_{ij}=0 otherwise. Each row has exactly d ones, so A_{n,d}
is a very sparse matrix, and it is real symmetric. Let
[
lambda_1ge lambda_2ge...ge lambda_n
]
be its eigenvalues, counted with multiplicity. They are real and lie in
-$[-d,d]$.
+[-d,d].
-A dense $ntimes n$ Gaussian matrix, by contrast, has all $tfrac12 n(n+1)$
+A dense ntimes n Gaussian matrix, by contrast, has all tfrac12 n(n+1)
entries random. Its eigenvalues are completely different objects at the
macroscopic scale. And yet, as we now explain, near any fixed interior
energy the two matrices produce the same local eigenvalue
@@ -59,28 +59,28 @@ statistics. That is the surprise.
Kesten (1959) and McKay (1981) computed the limiting eigenvalue density
-of $G_{n,d}$: the empirical measure
-$frac1nsum_{i=1}^ndelta_{lambda_i}$ converges weakly, in probability,
+of G_{n,d}: the empirical measure
+frac1nsum_{i=1}^ndelta_{lambda_i} converges weakly, in probability,
to the Kesten--McKay density
rho_d(t);=;frac{d,√(4(d-1)-t^2)}{2pi,(d^2-t^2)}
qquadfor |t|<2√(d-1),
-and $rho_d(t)=0$ otherwise. It integrates to $1$. Its support is the
-compact interval $[-2√(d-1),,2√(d-1)]$, the spectrum of the
-$d$-regular tree.
+and rho_d(t)=0 otherwise. It integrates to 1. Its support is the
+compact interval [-2√(d-1),,2√(d-1)], the spectrum of the
+d-regular tree.
-For example, when $d=3$ the support is $[-2sqrt2,2sqrt2]$ and
-$rho_3(0)=3sqrt8/(2picdot9)approx 0.150$, so the mean gap between
-neighbouring eigenvalues near $0$ is about $1/0.150approx 6.7$ in the
-variable $lambda$.
+For example, when d=3 the support is [-2sqrt2,2sqrt2] and
+rho_3(0)=3sqrt8/(2picdot9)approx 0.150, so the mean gap between
+neighbouring eigenvalues near 0 is about 1/0.150approx 6.7 in the
+variable lambda.
-So near a fixed energy $E$ with $|E|<2√(d-1)$ there are about
-$nrho_d(E)$ eigenvalues in a window of fixed length, hence gaps of
-order $1/n$. Macroscopic statements such as (eq:km-density) cannot
-distinguish two ensembles whose gaps differ by a factor of $2$; they
+So near a fixed energy E with |E|<2√(d-1) there are about
+nrho_d(E) eigenvalues in a window of fixed length, hence gaps of
+order 1/n. Macroscopic statements such as (eq:km-density) cannot
+distinguish two ensembles whose gaps differ by a factor of 2; they
only see averages over windows containing millions of eigenvalues. The
-interesting question is what happens at scale $1/n$, where individual
+interesting question is what happens at scale 1/n, where individual
eigenvalues are visible.
@@ -88,50 +88,50 @@ Fix once and for all an energy
[
Ein(-2√(d-1),,2√(d-1)),
]
-and write $rho=rho_d(E)>0$. Two positive numbers measure everything:
-the point $E$ and the density $rho$. The two-step zoom is
+and write rho=rho_d(E)>0. Two positive numbers measure everything:
+the point E and the density rho. The two-step zoom is
[
lambda_i;⟼; x_i:=n(lambda_i-E)
- qquad(spectral coordinates),
+ (spectral coordinates),
]
-and then, finally, we multiply by $rho$ to make the density at the
-origin equal to $1$:
+and then, finally, we multiply by rho to make the density at the
+origin equal to 1:
[
Xi_{n,d,E};:=;sum_{i=1}^n
delta_{,rho n(lambda_i-E)}.
]
A point measure is a locally finite sum of unit point masses, one
-per eigenvalue. The prefactor $rho n$ is chosen so that near $E$ the
-mean number of points per unit length of $Xi_{n,d,E}$ tends to $1$:
+per eigenvalue. The prefactor rho n is chosen so that near E the
+mean number of points per unit length of Xi_{n,d,E} tends to 1:
neighbouring eigenvalues sit about distance one apart in the new
coordinate.
-A point measure can be tested against a continuous function $h$ with
-compact support: the pairing is $int h,ddXi=sum_ih(points of Xi)$.
+A point measure can be tested against a continuous function h with
+compact support: the pairing is int h,ddXi=sum_ih(points of Xi).
Two random point measures have the same law exactly when
[
h;⟼;EEexp(-int h,ddXi)
]
is the same function for every bounded continuous compactly supported
-$hge0$. These are the Laplace functionals; they are the natural
+hge0. These are the Laplace functionals; they are the natural
notion of convergence of point processes, and they are exactly what
Theorem~thm:main compares.
-The comparison object is the Gaussian Orthogonal Ensemble. Let $H_n$ be
+The comparison object is the Gaussian Orthogonal Ensemble. Let H_n be
a real symmetric random matrix whose entries on and above the diagonal
-are independent centred Gaussians, with variance $1/n$ off the diagonal
-and $2/n$ on it. Its eigenvalues $mu_i$ follow the semicircle density
-$frac1{2pi}√(4-t^2)$ on $[-2,2]$, whose value at the origin is
-$1/pi$. Following the same zoom-and-unfold recipe, define
+are independent centred Gaussians, with variance 1/n off the diagonal
+and 2/n on it. Its eigenvalues mu_i follow the semicircle density
+frac1{2pi}√(4-t^2) on [-2,2], whose value at the origin is
+1/pi. Following the same zoom-and-unfold recipe, define
[
Xi_n^{GOE}:=sum_{i=1}^ndelta_{(n/pi)mu_i}.
]
[Main result]
-For every fixed integer $dge3$, every fixed energy $E$ in the open
-interval $(-2√(d-1),2√(d-1))$, and every nonnegative continuous
-compactly supported $h:RtoR$,
+For every fixed integer dge3, every fixed energy E in the open
+interval (-2√(d-1),2√(d-1)), and every nonnegative continuous
+compactly supported h:RtoR,
[
|
EEexp(-int h,ddXi_{n,d,E})
@@ -139,7 +139,7 @@ compactly supported $h:RtoR$,
EEexp(-int h,ddXi_n^{GOE})
|→ 0
]
-as $nto∞$ through all admissible sizes.
+as nto∞ through all admissible sizes.
Because the GOE point processes converge in law (Valk'o--Vir'ag), the
@@ -149,36 +149,34 @@ size comparable to the typical spacing, not only the one-point density,
not only energy-averaged correlations.
[In plain words]
-Take a random $d$-regular graph on a huge number $n$ of vertices. Stand
-at an energy $E$ inside the spectrum and zoom in on the eigenvalues
+Take a random d-regular graph on a huge number n of vertices. Stand
+at an energy E inside the spectrum and zoom in on the eigenvalues
there until the typical distance between neighbouring eigenvalues is
-$1$. The random picture of points you see has the same statistics as the
+1. The random picture of points you see has the same statistics as the
analogous picture for a dense random Gaussian matrix. This is true for
every interval, jointly, and it is happening although the two matrices
-have nothing superficially in common: one has only $dn/2$ random bits,
-the other has about $n^2/2$.
+have nothing superficially in common: one has only dn/2 random bits,
+the other has about n^2/2.
Two features of the statement deserve emphasis.
-
-
-item No conditioning. The theorem makes no restriction to
+ No conditioning. The theorem makes no restriction to
connected or non-bipartite graphs. The relevant local information is
present in the unconditioned ensemble, and events of vanishing
probability are discarded freely.
-item Fixed degree, fixed energy. Both $d$ and $E$ are held fixed
-while $nto∞$. Degree $d=2$ is genuinely different: a $2$-regular
+ Fixed degree, fixed energy. Both d and E are held fixed
+while nto∞. Degree d=2 is genuinely different: a 2-regular
graph is a disjoint union of cycles, all eigenvalues are explicit
cosines, and the local statistics are not the Gaussian ones. Thus
-$dge3$ is the honest hypothesis, and ``$d$ grows with $n$'' is a
+dge3 is the honest hypothesis, and ``d grows with n'' is a
different (earlier) theorem.
-A random $d$-regular graph looks locally like the infinite $d$-regular
+A random d-regular graph looks locally like the infinite d-regular
tree: for a fixed radius, the ball around a typical vertex is a tree
-of degree $d$. Trees are not Gaussian; their spectral measure is
-$rho_d$, not the semicircle. And yet Theorem~thm:main says the
+of degree d. Trees are not Gaussian; their spectral measure is
+rho_d, not the semicircle. And yet Theorem~thm:main says the
microscopic eigenvalue statistics agree with those of a dense
Gaussian matrix. Locality of the graph and universality of the
eigenvalues coexist.
@@ -187,33 +185,31 @@ The conjecture in this form was made by Jakobson, Miller, Rivin and
Rudnick (1999), who also gave numerical evidence; they phrased it in
terms of nearest-neighbour spacings, the older and less complete
formulation of the same physics. What was known before this result:
-
-
-item Macroscopic law. Kesten, McKay: (eq:km-density).
-item Local laws. Bauerschmidt--Huang--Yau and Huang--Yau proved
+ Macroscopic law. Kesten, McKay: (eq:km-density).
+ Local laws. Bauerschmidt--Huang--Yau and Huang--Yau proved
the local Kesten--McKay law, delocalization and rigidity for fixed
-$dge 3$; these are the spectral inputs used below. They are estimates
+dge 3; these are the spectral inputs used below. They are estimates
for smoothed eigenvalue counts (spectral windows of width at
-least $n^{-1+o(1)}$), much coarser than individual spacings.
-item Growing degree. Bauerschmidt--Huang--Knowles--Yau obtained
+least n^{-1+o(1)}), much coarser than individual spacings.
+ Growing degree. Bauerschmidt--Huang--Knowles--Yau obtained
bulk gap universality for polynomial degrees; Bourgade--Huang obtained
-fixed-energy point-process universality for $(log n)^{24}ll dle n^{1/2}$
+fixed-energy point-process universality for (log n)^{24}ll dle n^{1/2}
and stated the fixed-degree problem as their Conjecture 2.11.
-item Edges. Huang--McKenzie--Yau proved optimal rigidity and
-GOE edge universality for every fixed $dge3$.
-item Gaussian waves. Backhausz--Szegedy proved qualitative
+ Edges. Huang--McKenzie--Yau proved optimal rigidity and
+GOE edge universality for every fixed dge3.
+ Gaussian waves. Backhausz--Szegedy proved qualitative
Gaussian limits for ``almost eigenvectors''; the entropy framework of
Backhausz--Szegedy--Trees and Backhausz--Bordenave--Szegedy is the
engine we upgrade to a quantitative, variance-one statement.
-Two new mechanisms are needed for fixed $d$, and they are the heart of
+Two new mechanisms are needed for fixed d, and they are the heart of
the proof.
First, weak convergence of eigenvector statistics is not enough: we need
quantitative Gaussian moments, polynomially accurate, uniformly
for all eigenvectors in a spectral band. The entropy argument delivers
-exactly this, with an error $n^{-c}$ on an event of probability tending
+exactly this, with an error n^{-c} on an event of probability tending
to one, uniformly over graph-dependent choices of eigenvectors.
Second, the local laws know only about sums of resolvents over many
@@ -232,32 +228,27 @@ Dyson Brownian motion universality finishes the job.
Everything rests on four propositions, proved in order, plus a
collection of imported theorems (local laws, edge universality,
Dyson Brownian motion universality, semicircle edge limits).
-
-
-item the entropy argument: Gaussian moments at sampled
-edges. For fixed $d,E$ and any fixed number of eigenvectors in a
+ the entropy argument: Gaussian moments at sampled
+edges. For fixed d,E and any fixed number of eigenvectors in a
microscopic band, the joint values of those eigenvectors at random edge
endpoints are Gaussian to polynomial precision.
Mechanism: a colored-star counting inequality becomes a
differential-entropy inequality after smoothing by a Gaussian field;
then a strict tensor inequality turns the small entropy deficit into
moment matching, by induction over Hermite degree.
-
-item the microscopic-limit step: endpoint resolvents and the
+ the microscopic-limit step: endpoint resolvents and the
microscopic limit. The moments upgrade to control of resolvent
compressions at sampled edges, uniformly down to fixed microscopic
height. A two-edge switch gives the scalar self-consistency equation
that pins the trace. Then a subsequential limit of the microscopic
process exists and comes with Gaussian endpoint marks.
-
-item the switching step: continuous switching and
+ the switching step: continuous switching and
Gaussian insertion. The discrete switch symmetry, combined with
orthogonal invariance of the Gaussian marks, becomes an exact
continuous symmetry of the limiting law. Composing many small
orthogonal switches produces, in the limit, a diagonal matrix plus a
GOE perturbation with the same point process as the microscopic law.
-
-item the Dyson Brownian motion step: identification. The diagonal
+ the Dyson Brownian motion step: identification. The diagonal
matrices satisfy the hypotheses of Dyson Brownian motion universality
at a single energy, after an exact adjustment of the free-convolution
density. A uniform exponential bound on GOE counts upgrades correlation
@@ -276,15 +267,15 @@ is produced by the entropy step and then transported by symmetry and
Brownian motion.
-Throughout, $dge3$ and $E$ with $|E|<2√(d-1)$ are fixed, and every
-constant may depend on $d$ and $E$ without comment. We write $c,C$ for
+Throughout, dge3 and E with |E|<2√(d-1) are fixed, and every
+constant may depend on d and E without comment. We write c,C for
positive constants whose values may change from one occurrence to the
-next, and $o(1)$ for a quantity tending to zero as $nto∞$. On
-events of probability tending to one we write $n^{o(1)}$ for a
-deterministic subpolynomial bound. The notation $O_{PP}(b)$ means a
-random quantity whose ratio to $b$ is bounded in probability. All
-eigenvectors are real eigenbasis vectors of $A_{n,d}$, normalized to
-have squared norm $n$; the choice inside an eigenspace is arbitrary
+next, and o(1) for a quantity tending to zero as nto∞. On
+events of probability tending to one we write n^{o(1)} for a
+deterministic subpolynomial bound. The notation O_{PP}(b) means a
+random quantity whose ratio to b is bounded in probability. All
+eigenvectors are real eigenbasis vectors of A_{n,d}, normalized to
+have squared norm n; the choice inside an eigenspace is arbitrary
and measurable.
@@ -296,78 +287,78 @@ spectral estimates are quoted; everything else here is proved.
[In plain words]
Three pieces of background are fixed here once and for all. First, the
-uniform $d$-regular graph is studied through the configuration model:
-$d$ labelled stubs at every vertex, matched uniformly at random, then
+uniform d-regular graph is studied through the configuration model:
+d labelled stubs at every vertex, matched uniformly at random, then
conditioned on producing a simple graph; the conditioning is uniform,
with probability bounded below. Second, the sharp local spectral
estimates of Huang--Yau give rigidity, delocalization, and a resolvent
-approximation by the infinite tree value at every height above $S/n$.
-Third, on the infinite $d$-regular tree, boundary values of the resolvent
+approximation by the infinite tree value at every height above S/n.
+Third, on the infinite d-regular tree, boundary values of the resolvent
give the exact covariance of eigenvector values at nearby vertices; this
is what makes the Gaussian endpoint marks appear later.
-For $Imzeta>0$ put
+For Imzeta>0 put
[
m_d(zeta)=int_{R}(rho_d(t))/(t-zeta),dd t,
]
the Stieltjes transform of the Kesten--McKay density. It is analytic in
-the upper half-plane with $Im m_d>0$. The boundary value from above
-exists and has imaginary part $pirho_d$ on the open support. Unless a
+the upper half-plane with Im m_d>0. The boundary value from above
+exists and has imaginary part pirho_d on the open support. Unless a
subscript says otherwise we set, once and for all,
- rho=rho_d(E),qquad p=frac Ed,qquad
- a=(d)/(d-1)(1-p^2),qquad
+ rho=rho_d(E), p=frac Ed,
+ a=(d)/(d-1)(1-p^2),
m_*=m_d(E+mathrm i0)=m_rmi0)=m_{rm r}+mathrm irmipirho .
-Then $rho>0$, $a>0$, $pin(-1,1)$, and $m_{rm r}$ is a real number.
+Then rho>0, a>0, pin(-1,1), and m_{rm r} is a real number.
Note that
- 1-p^2=(d^2-E^2)/(d^2),qquad
+ 1-p^2=(d^2-E^2)/(d^2),
a=(d^2-E^2)/(d(d-1))=(d)/(d-1)(1-p^2).
-Beware that $a$ and $1-p^2$ differ by the factor $d/(d-1)$; both forms
+Beware that a and 1-p^2 differ by the factor d/(d-1); both forms
are used below and they are not interchangeable.
-Let $u_1,...,u_n$ be a real orthonormal eigenbasis of $A=A_{n,d}$,
-normalized by $|u_i|^2=n$, so $Au_i=lambda_iu_i$ and
-$< u_i,u_j>=ndelta_{ij}$. Define the microscopic
+Let u_1,...,u_n be a real orthonormal eigenbasis of A=A_{n,d},
+normalized by |u_i|^2=n, so Au_i=lambda_iu_i and
+< u_i,u_j>=ndelta_{ij}. Define the microscopic
coordinates and the empirical spectral measure of the zoomed picture:
- x_i=n(lambda_i-E),qquad
- nu_n=sum_{i=1}^ndelta_{x_i},qquad
+ x_i=n(lambda_i-E),
+ nu_n=sum_{i=1}^ndelta_{x_i},
m_n(z)&=sum_{i=1}^nfrac1{x_i-z}
=frac1nTr R_n(z),
- R_n(z)&=(A-E-frac zn)^{-1},quad Im z>0 .
+ R_n(z)&=(A-E-frac zn)^{-1}, Im z>0 .
-Thus $m_n$ is the Stieltjes transform of $nu_n$ in the coordinates
-$x_i$, and $nu_n$ is recovered from $m_n$ by the Stieltjes inversion
+Thus m_n is the Stieltjes transform of nu_n in the coordinates
+x_i, and nu_n is recovered from m_n by the Stieltjes inversion
formula. The final unfolded coordinates in
-Theorem~thm:main are obtained from the $x_i$ by one more
-multiplication by $rho$; this final scaling is a deterministic
+Theorem~thm:main are obtained from the x_i by one more
+multiplication by rho; this final scaling is a deterministic
continuous operation and will be applied only at the very end.
-For $kge1$ define
+For kge1 define
- J_k=0&I_k I_k&0,qquad
- C_k=I_{2k}+pJ_k,qquad
+ J_k=0&I_k I_k&0,
+ C_k=I_{2k}+pJ_k,
K_k(m)=mC_k+d^{-1}J_k .
-The ordering of the $2k$ coordinates is always: the $k$ starting
-vertices of $k$ sampled oriented edges, then their $k$ ending vertices.
-This ordering matters and is kept fixed. The eigenvalues of $J_k$ are
-$+1$ and $-1$, each with multiplicity $k$, so the eigenvalues of $C_k$
-are $1+p$ and $1-p$: both positive because $|p|<1$. In particular
-$C_k$ is positive definite and $C_k^{1/2}$ is well defined and bounded
-with bounded inverse, uniformly in $k$.
+The ordering of the 2k coordinates is always: the k starting
+vertices of k sampled oriented edges, then their k ending vertices.
+This ordering matters and is kept fixed. The eigenvalues of J_k are
++1 and -1, each with multiplicity k, so the eigenvalues of C_k
+are 1+p and 1-p: both positive because |p|<1. In particular
+C_k is positive definite and C_k^{1/2} is well defined and bounded
+with bounded inverse, uniformly in k.
-The matrix $C_k$ is the covariance we will find for the Gaussian marks:
+The matrix C_k is the covariance we will find for the Gaussian marks:
the value of an eigenvector at a sampled vertex has variance one after
-normalization, and the two ends of a sampled edge have covariance $p$.
+normalization, and the two ends of a sampled edge have covariance p.
We take graphs from the configuration (pairing) model and condition on
@@ -375,31 +366,31 @@ simplicity. The facts we need are elementary and are proved here rather
than quoted.
-In the configuration model with $d$ labelled half-edges, or stubs, at
-each of $n$ labelled vertices, and with a uniformly random matching of
-the $nd$ stubs, conditioning the matching to produce a simple graph
-gives exactly the uniform simple labelled $d$-regular graph $G_{n,d}$.
-Moreover, along all admissible $n$,
+In the configuration model with d labelled half-edges, or stubs, at
+each of n labelled vertices, and with a uniformly random matching of
+the nd stubs, conditioning the matching to produce a simple graph
+gives exactly the uniform simple labelled d-regular graph G_{n,d}.
+Moreover, along all admissible n,
[
PP(the matching produces a simple graph)
→ exp(-(d^2-1)/(4))>0 .
]
-In a uniform simple $d$-regular graph: a fixed vertex lies in a triangle
-with probability $O(n^{-1})$; and, for any fixed $k$, $k$
+In a uniform simple d-regular graph: a fixed vertex lies in a triangle
+with probability O(n^{-1}); and, for any fixed k, k
independently sampled uniform oriented edges have distinct endpoints
and induce exactly the sampled matching, with probability
-$1-O_k(n^{-1})$.
+1-O_k(n^{-1}).
Proof.
-Each simple $d$-regular graph corresponds to exactly $(d!)^n$ matchings
-(one for each ordering of the stubs at the $n$ vertices), so conditioning
+Each simple d-regular graph corresponds to exactly (d!)^n matchings
+(one for each ordering of the stubs at the n vertices), so conditioning
on simplicity is uniform.
For the asymptotic probability of being simple, count the expected
number of defects. A loop is a matched pair of stubs at the same
-vertex; there are $binom d2$ candidate pairs at each of $n$ vertices,
-each realized with probability $1/(nd-1)$. So the expected number of
+vertex; there are binom d2 candidate pairs at each of n vertices,
+each realized with probability 1/(nd-1). So the expected number of
loops is
[
n,(d(d-1))/(2)cdotfrac1{nd-1}→ (d-1)/(2).
@@ -412,31 +403,31 @@ edges is
binom n2cdot ((binom d2)^2cdot 2)/((nd-1)(nd-3))
→ ((d-1)^2)/(4).
]
-A prescribed family of $e$ disjoint stub pairs is realized with
-probability $prod_{j=0}^{e-1}(nd-1-2j)^{-1}sim(nd)^{-e}$, for fixed
-$e$. Hence, for fixed $j$, the $j$-th factorial moment of the total
-defect count $Z$ (loops plus double edges) is asymptotic to
-$((d-1)/(2)+((d-1)^2)/(4))^j$: for disjoint defect
+A prescribed family of e disjoint stub pairs is realized with
+probability prod_{j=0}^{e-1}(nd-1-2j)^{-1}sim(nd)^{-e}, for fixed
+e. Hence, for fixed j, the j-th factorial moment of the total
+defect count Z (loops plus double edges) is asymptotic to
+((d-1)/(2)+((d-1)^2)/(4))^j: for disjoint defect
tuples this is immediate from the product formula, and for tuples that
-share vertices one checks that they contribute $o(1)$. Indeed, view the
-prescribed pairs of an overlapping tuple as a multigraph with $v$
-vertices and $e$ edges. Every vertex has degree at least $2$. A
-connected component with all degrees exactly $2$ is a single prescribed
+share vertices one checks that they contribute o(1). Indeed, view the
+prescribed pairs of an overlapping tuple as a multigraph with v
+vertices and e edges. Every vertex has degree at least 2. A
+connected component with all degrees exactly 2 is a single prescribed
loop or a single prescribed double edge; a genuinely overlapping
-component has $v<e$, there are $O(n^v)$ choices, and each occurs with
-probability $O(n^{-e})$. Since there are only finitely many overlap
-patterns for fixed $d,j$, the contribution is $o(1)$. Since $Z$
+component has v<e, there are O(n^v) choices, and each occurs with
+probability O(n^{-e}). Since there are only finitely many overlap
+patterns for fixed d,j, the contribution is o(1). Since Z
converges in distribution to a Poisson variable with parameter
-$(d^2-1)/(4)$, and $Z=0$ is exactly simplicity, we get the stated
+(d^2-1)/(4), and Z=0 is exactly simplicity, we get the stated
limit; in particular it is positive.
For a triangle through a fixed vertex, sum over the other two vertices
-and their stub choices: $O(n^2)cdot O(n^{-3})=O(n^{-1})$ in the
+and their stub choices: O(n^2)cdot O(n^{-3})=O(n^{-1}) in the
matching model, and conditioning on simplicity multiplies this by a
bounded factor.
-Finally, in a simple $d$-regular graph a uniform vertex is uniform. Each
-previously chosen endpoint has only $O_d(1)$ vertices at distance at
+Finally, in a simple d-regular graph a uniform vertex is uniform. Each
+previously chosen endpoint has only O_d(1) vertices at distance at
most one from it, and a uniform oriented edge has a uniform starting
vertex. A union bound over the fixed number of sampled edges gives the
assertion.
@@ -447,65 +438,64 @@ Put
S=exp(√(log n)),
-a quantity that is $n^{o(1)}$ but larger than any fixed power of
-$log n$. We import the following deliberately weak package from
+a quantity that is n^{o(1)} but larger than any fixed power of
+log n. We import the following deliberately weak package from
Huang--Yau.
[Spectral input]
-There exist constants $c_0,c_1>0$, $B<∞$, and an open interval
-$I_E$ containing $E$ whose closure lies in
-$(-2√(d-1),2√(d-1))$, such that the following hold with
-probability tending to one, along all admissible $n$.
-
-item After deleting the eigenvalue attached to the constant vector, the
-eigenvalues are within $n^{-c_0}$ of their Kesten--McKay classical
+There exist constants c_0,c_1>0, B<∞, and an open interval
+I_E containing E whose closure lies in
+(-2√(d-1),2√(d-1)), such that the following hold with
+probability tending to one, along all admissible n.
+ After deleting the eigenvalue attached to the constant vector, the
+eigenvalues are within n^{-c_0} of their Kesten--McKay classical
locations; and every eigenvector entry is bounded by a deterministic
-$n^{o(1)}$.
-item Uniformly for $Rezetain I_E$ and $S/nleImzetale 1$,
+n^{o(1)}.
+ Uniformly for Rezetain I_E and S/nleImzetale 1,
[
|frac1nTr(A-zeta)^{-1}-m_d(zeta)|
le (log n)^B(n^{-c_1}+(nImzeta)^{-c_1}).
]
-Except possibly at $n^{1-c_1}$ centres, the resolvent entries at a
-centre and at its $d$ neighbours agree with the corresponding infinite
-$d$-regular-tree entries up to the same bound.
+Except possibly at n^{1-c_1} centres, the resolvent entries at a
+centre and at its d neighbours agree with the corresponding infinite
+d-regular-tree entries up to the same bound.
-In particular $#{i:|x_i|le S^2}=O(S^2)$ with probability tending to
+In particular #{i:|x_i|le S^2}=O(S^2) with probability tending to
one.
Proof.
This is a restatement in our normalization of Theorems 1.2, 1.4 and 4.2
of Huang--Yau, in the arXiv version 2102.00963v3. Their matrix is
-$A/√(d-1)$, so their bulk energy is $E/√(d-1)in(-2,2)$;
+A/√(d-1), so their bulk energy is E/√(d-1)in(-2,2);
rescaling the spectral parameter and the resolvent by the fixed factor
-$√(d-1)$ gives our normalization.
+√(d-1) gives our normalization.
We keep track of why their error has our shape. Choose fixed
-$0<c<1$, $a_{rm HY}ge12$, $b_{rm HY}ge25a_{rm HY}$ and
-$0<mathfrak r<c/32$,frakr<c/32$, and let $r_{rm HY}$ be
-$mathfrak rfrakrlog_{d-1}n$ rounded to an integer. Their Definition 1.1
+0<c<1, a_{rm HY}ge12, b_{rm HY}ge25a_{rm HY} and
+0<mathfrak r<c/32,frakr<c/32, and let r_{rm HY} be
+mathfrak rfrakrlog_{d-1}n rounded to an integer. Their Definition 1.1
and Proposition 2.1 produce an event of probability tending to one on
which all but a polynomially small fraction of vertices have a tree
-neighbourhood of radius $(c/4)log_{d-1}n$; that radius is much larger
-than $r_{rm HY}$, and in particular larger than any neighbourhood
+neighbourhood of radius (c/4)log_{d-1}n; that radius is much larger
+than r_{rm HY}, and in particular larger than any neighbourhood
appearing below. On that event their resolvent comparison, with their
error (4.6)--(4.7), bounds the discrepancy by a logarithmic factor
times
[
(d-1)^{-r_{rm HY}}+(nImzeta)^{-1/2}+(nImzeta)^{-2/3}.
]
-Since $S$ dominates every fixed power of $log n$, and after decreasing
-$c_1$ and increasing $B$, this is the stated bound, uniform in the
+Since S dominates every fixed power of log n, and after decreasing
+c_1 and increasing B, this is the stated bound, uniform in the
indicated range. The same two statements supply polynomial rigidity and
polylogarithmic complete delocalization, which is the bound on
eigenvector entries.
-For the count, apply the trace estimate at $zeta=E+mathrm irmi S^2/n$.
-Every eigenvalue with $|lambda_i-E|le S^2/n$ contributes at least
-$n/(2S^2)$ to the imaginary part of $(x_i-zeta/n)^{-1}$; the
+For the count, apply the trace estimate at zeta=E+mathrm irmi S^2/n.
+Every eigenvalue with |lambda_i-E|le S^2/n contributes at least
+n/(2S^2) to the imaginary part of (x_i-zeta/n)^{-1}; the
normalized imaginary trace is bounded by the estimator above, and the
-count is $O(S^2)$.
+count is O(S^2).
@@ -523,26 +513,26 @@ records how eigenvector values at nearby vertices are correlated. We
derive it, together with the logarithmic potential that appears in the
entropy computation, so that no normalization is hidden.
-Fix a locally finite graph with adjacency matrix $A$ and a point $E$.
-For vertices $v,w$ define the covariance kernel from the boundary
-values of $G^zeta=(A-zeta)^{-1}$:
+Fix a locally finite graph with adjacency matrix A and a point E.
+For vertices v,w define the covariance kernel from the boundary
+values of G^zeta=(A-zeta)^{-1}:
K(v,w)=lim_{eta↓0}frac{Im< e_v,(A-E-ieta)^{-1}e_w>}{pirho}.
-The kernel in the numerator, $eta/((A-E)^2+eta^2)$, is
-positive semidefinite, so $K$ is a covariance kernel.
+The kernel in the numerator, eta/((A-E)^2+eta^2), is
+positive semidefinite, so K is a covariance kernel.
-Let $q(zeta)$ be the solution of
+Let q(zeta) be the solution of
[
zeta=q^{-1}+(d-1)q
]
-that tends to $0$ as $zetato∞$ in the upper half-plane. Then
+that tends to 0 as zetato∞ in the upper half-plane. Then
-m_d(zeta)=frac1{zeta-dq(zeta)} .
The covariance kernel (eq:cov-definition) of the infinite
-$d$-regular tree is: variance $1$ at every vertex, covariance $p$ between
+d-regular tree is: variance 1 at every vertex, covariance p between
a centre and a neighbour, and covariance
[
(E^2-d)/(d(d-1))
@@ -552,40 +542,40 @@ between two distinct neighbours of a centre. Moreover
k_0=(1+Em_*)/(d)
quadsatisfiesquad k_0^2-k_0-m_*^2=0 .
-Finally, with $U(E)=intlog|t-E|rho_d(t),dd t$,
+Finally, with U(E)=intlog|t-E|rho_d(t),dd t,
-U(E)=-frac12log d+(d-1)/(2)log a-frac d4log(1-p^2),
-and replacing $E$ by $E+mathrm irmieta$ changes the real part of the
-logarithmic potential by $O(eta)$.
+and replacing E by E+mathrm irmieta changes the real part of the
+logarithmic potential by O(eta).
Proof.
-The Green function. Let $o$ be the root of the full tree and let
-$F(zeta)$ be the diagonal entry of $(zeta-A_{rm branch})^{-1}$ at the
-root of a rooted tree in which the root has $d-1$ children, recursively.
+The Green function. Let o be the root of the full tree and let
+F(zeta) be the diagonal entry of (zeta-A_{rm branch})^{-1} at the
+root of a rooted tree in which the root has d-1 children, recursively.
Each child subtree is again such a branch, so a Schur complement at the
root gives
[
F=frac1{zeta-(d-1)F},
- qquadi.e.qquad zeta=F^{-1}+(d-1)F,
+ qquadi.e. zeta=F^{-1}+(d-1)F,
]
-and $Fto0$ at infinity; thus $F=q$. At the root of the full tree, a
-Schur complement over the $d$ branches gives
+and Fto0 at infinity; thus F=q. At the root of the full tree, a
+Schur complement over the d branches gives
[
bigl< e_o,(zeta-A)^{-1}e_obigr>=frac1{zeta-dq}.
]
-The left side is $intrho_d(t)/(zeta-t)dd t=-m_d(zeta)$, which proves
+The left side is intrho_d(t)/(zeta-t)dd t=-m_d(zeta), which proves
(eq:tree-green). Its boundary imaginary part is
-$pi^{-1}Im m_d(t+mathrm i0)=rmi0)=rho_d(t)$: this follows by checking that
+pi^{-1}Im m_d(t+mathrm i0)=rmi0)=rho_d(t): this follows by checking that
the boundary function is nonnegative, compactly supported on
-$[-2√(d-1),2√(d-1)]$, and has total mass $1$, the latter from
-$m_d(zeta)sim-1/zeta$ at infinity and a contour argument.
+[-2√(d-1),2√(d-1)], and has total mass 1, the latter from
+m_d(zeta)sim-1/zeta at infinity and a contour argument.
-Covariance. Write $G=(A-zeta)^{-1}$ on the tree and abbreviate
-$g=G_{oo}=m_d$, $g_1=G_{ox}$ for a neighbour $x$, $g_2=G_{xy}$ for two
-distinct neighbours of $o$; these are independent of the chosen
-vertices by homogeneity. The resolvent equation $(zeta-A)G=I$ at the
+Covariance. Write G=(A-zeta)^{-1} on the tree and abbreviate
+g=G_{oo}=m_d, g_1=G_{ox} for a neighbour x, g_2=G_{xy} for two
+distinct neighbours of o; these are independent of the chosen
+vertices by homogeneity. The resolvent equation (zeta-A)G=I at the
root gives
[
zeta g-1=d g_1 ,
@@ -595,76 +585,76 @@ and applied to a neighbour column gives
[
zeta g_1=g+(d-1)g_2 .
]
-Taking $zeta↓ E$, the imaginary parts of $g=pirho$ and of
-$g_1=ppirho$ give, after dividing by $pirho$, the variance $1$ and
-the centre-neighbour covariance $p$; then
+Taking zeta↓ E, the imaginary parts of g=pirho and of
+g_1=ppirho give, after dividing by pirho, the variance 1 and
+the centre-neighbour covariance p; then
[
(d-1)Im g_2=Im(zeta g_1)-Im g
=(E^2)/(d)pirho-pirho,
]
-i.e. $Im g_2/(pirho)=(E^2-d)/(d(d-1))$, which is the stated
+i.e. Im g_2/(pirho)=(E^2-d)/(d(d-1)), which is the stated
distinct-neighbour covariance. By (eq:tree-green) the function
-$q$ and hence $m_d$ are analytic across each side of a small complex
-neighbourhood of $E$, so all these boundary values are attained with
+q and hence m_d are analytic across each side of a small complex
+neighbourhood of E, so all these boundary values are attained with
polynomial rate in the imaginary part.
-The quadratic equation. Attach the value $q$ at $zeta=E+i0$ to
-the identity $zeta-dq=-1/m$; eliminating $q$ using
-$zeta q-(d-1)q^2=1$ gives the quadratic
+The quadratic equation. Attach the value q at zeta=E+i0 to
+the identity zeta-dq=-1/m; eliminating q using
+zeta q-(d-1)q^2=1 gives the quadratic
[
(d^2-E^2)m^2+E(d-2)m+(d-1)=0quadat m=m_* .
]
-The same computation with $m$ arbitrary is exactly
-$k_0^2-k_0-m^2=0$ for $k_0=(1+Em)/d$, since
+The same computation with m arbitrary is exactly
+k_0^2-k_0-m^2=0 for k_0=(1+Em)/d, since
[
((1+Em)/(d))^2-(1+Em)/(d)-m^2
=((E^2-d^2)m^2+E(2-d)m+(1-d))/(d^2),
]
which is a negative multiple of the left side above. This is
(eq:k0). In the strict bulk the quadratic has two real roots, one
-of which is $m_*$ with $Im m_*=+pirho$.
+of which is m_* with Im m_*=+pirho.
The potential. Differentiate the analytic function
-$-m_d(zeta)=q/(1-q^2)$ and integrate along a path with
-$zeta=q^{-1}+(d-1)q$:
+-m_d(zeta)=q/(1-q^2) and integrate along a path with
+zeta=q^{-1}+(d-1)q:
[
Reintlog(zeta-t),rho_d(t),dd t
=-log|q|-(d-2)/(2)log|1-q^2| ,
]
-matched at infinity using $m_d(zeta)sim-1/zeta$. At the boundary,
+matched at infinity using m_d(zeta)sim-1/zeta. At the boundary,
[
- |q|=(d-1)^{-1/2},qquad
+ |q|=(d-1)^{-1/2},
|1-q^2|=frac{√(d^2-E^2)}{d-1} ,
]
-by evaluating the quadratic at $zeta=E$. Substituting into
-(eq:tree-potential) and using $a=d(1-p^2)/(d-1)$ in the form
-$1-p^2=(d^2-E^2)/d^2$ gives the formula. (As a check at $d=3$, $E=0$:
-$|q|=1/sqrt2$, $|1-q^2|=3/2$, and the right side is
-$log(sqrt3/2)$.)
+by evaluating the quadratic at zeta=E. Substituting into
+(eq:tree-potential) and using a=d(1-p^2)/(d-1) in the form
+1-p^2=(d^2-E^2)/d^2 gives the formula. (As a check at d=3, E=0:
+|q|=1/sqrt2, |1-q^2|=3/2, and the right side is
+log(sqrt3/2).)
-Finally, $tmapstolog|t-E|$ is bounded near $E$ because $rho_d$ is,
+Finally, tmapstolog|t-E| is bounded near E because rho_d is,
and
[
intlog|1-(mathrm irmieta)/(t-E)|rho_d(t),dd t
=frac12intlog(1+(eta^2)/((t-E)^2))rho_d(t),dd t
=O(eta),
]
-by the substitution $t-E=eta u$ and the integrability of
-$log(1+u^{-2})$ at infinity.
+by the substitution t-E=eta u and the integrability of
+log(1+u^{-2}) at infinity.
The first of the four steps gives a fully quantitative answer to a
qualitative question: do eigenvector values look Gaussian? The answer is
yes, with polynomial accuracy, uniformly over all choices of eigenvectors
-in a microscopic band around $E$. This is the only place where the graph
+in a microscopic band around E. This is the only place where the graph
structure (as opposed to imported spectral input) is used in the first
half of the proof.
Why do we need such precision? A fixed moment of a fixed number of
eigenvectors is later summed over eigenindices, and the number of
-indices in a band of width $S^2/n$ is $n^{o(1)}$; a fixed negative power
-$n^{-c}$ is more than enough to beat any subpolynomial factor. What
+indices in a band of width S^2/n is n^{o(1)}; a fixed negative power
+n^{-c} is more than enough to beat any subpolynomial factor. What
would not be enough is abstract weak convergence, which carries no rate.
The strategy is the entropy method of Backhausz--Szegedy, made
@@ -681,40 +671,40 @@ behaviour.
[Uniform endpoint moments]
-Fix positive integers $k,ell$ and a nonnegative integer $q$. There are
-constants $c,C>0$, and events of graphs with probability tending to one,
+Fix positive integers k,ell and a nonnegative integer q. There are
+constants c,C>0, and events of graphs with probability tending to one,
with the following property. Choose distinct indices
-$i(1),...,i(ell)$ satisfying $|x_{i(j)}|le S^2$, by any rule that
-depends only on the graph. Independently of that choice, sample $k$
-uniform oriented edges $(v_1,v'_1),...,(v_k,v'_k)$. Set
+i(1),...,i(ell) satisfying |x_{i(j)}|le S^2, by any rule that
+depends only on the graph. Independently of that choice, sample k
+uniform oriented edges (v_1,v'_1),...,(v_k,v'_k). Set
[
b_j=(u_{i(j)}(v_1),...,u_{i(j)}(v_k),
u_{i(j)}(v'_1),...,u_{i(j)}(v'_k))^{mathsf T}sfT
inR^{2k},
- qquad j=1,...,ell .
+ j=1,...,ell .
]
-Then for every monomial $P$ of total degree at most $q$ in the
-coordinates of $b_1,...,b_ell$,
+Then for every monomial P of total degree at most q in the
+coordinates of b_1,...,b_ell,
|EE[P(b_1,...,b_ell)mid A]
-EE P(Z_1,...,Z_ell)|le Cn^{-c},
-where $Z_1,...,Z_ell$ are independent centred Gaussian vectors with
-covariance $C_k$ from (eq:channel-matrices).
+where Z_1,...,Z_ell are independent centred Gaussian vectors with
+covariance C_k from (eq:channel-matrices).
The constants and the graph events are uniform over the choices of
-eigenindices. They depend on $d,E,k,ell,q$ but not on $n$; no assertion
-is made for dimensions or moment orders that grow with $n$. The
-covariance $C_k$ is the one predicted by the tree: variance $1$, and
-covariance $p$ across each sampled edge, with zero covariance between
+eigenindices. They depend on d,E,k,ell,q but not on n; no assertion
+is made for dimensions or moment orders that grow with n. The
+covariance C_k is the one predicted by the tree: variance 1, and
+covariance p across each sampled edge, with zero covariance between
different edges.
[In plain words]
Choose finitely many eigenvectors whose eigenvalues sit within
-$S^2/n$ of $E$. Look at the values of these eigenvectors at the two
-ends of a few random edges. Those values behave, to accuracy $n^{-c}$
-and with probability $1-o(1)$, like independent Gaussian vectors whose
+S^2/n of E. Look at the values of these eigenvectors at the two
+ends of a few random edges. Those values behave, to accuracy n^{-c}
+and with probability 1-o(1), like independent Gaussian vectors whose
covariance matrix is the tree covariance. ``Being Gaussian'' here means
every fixed polynomial moment matches, which is enough for everything
we do later.
@@ -727,18 +717,18 @@ Write
]
for the vector of the chosen eigenvectors at a vertex. A star
observation is the value at a uniform vertex together with the values
-at its $d$ neighbours in uniformly random order; an edge
+at its d neighbours in uniformly random order; an edge
observation is the pair of values at the two ends of a uniform oriented
edge. All randomness in an observation includes the choice of location;
the graph is conditioned on.
We need two notions of entropy. For a random variable on a finite set we
-write $H$ for the discrete entropy; for a random vector with a density
-we write $mathcal H$calH$ for the differential entropy; both use natural
-logarithms. The only fact about $H$ we use besides its definition is
+write H for the discrete entropy; for a random vector with a density
+we write mathcal HcalH for the differential entropy; both use natural
+logarithms. The only fact about H we use besides its definition is
that the number of labelings with a given empirical distribution of
-local observations is at most exponential in $n$ times the entropy of
-that distribution; the only facts about $mathcal H$calH$ are the Gaussian
+local observations is at most exponential in n times the entropy of
+that distribution; the only facts about mathcal HcalH are the Gaussian
maximum-entropy bound and the continuity estimates for binning, both
spelled out below.
@@ -747,72 +737,72 @@ the next; this is harmless because only finitely many steps occur.
The first ingredient does not involve eigenvectors at all. Give the
-vertices of the configuration model labels from an alphabet of size $M$.
-Let $T$ be the distribution of the ordered star observation, symmetrized
-over the $d!$ orders of the neighbours, and let $Q$ be the distribution
-of the oriented edge observation. Then $T$ and $Q$ are determined by the
+vertices of the configuration model labels from an alphabet of size M.
+Let T be the distribution of the ordered star observation, symmetrized
+over the d! orders of the neighbours, and let Q be the distribution
+of the oriented edge observation. Then T and Q are determined by the
labeling, and they are the only local information the counting
inequality sees. The following is the coloured-star count of
Backhausz--Szegedy, with the dependence on a growing alphabet kept
explicit.
[Uniform type count]
-Suppose $M=M_n$ and, for some fixed $kappa>0$,
+Suppose M=M_n and, for some fixed kappa>0,
M^{d+1}log nle n^{1-kappa}.
With probability tending to one, simultaneously for every pair of
-histograms $T,Q$, the number of vertex labelings giving these histograms
+histograms T,Q, the number of vertex labelings giving these histograms
is at most
expBigl{nH(T)-(nd)/(2)H(Q)+O(n^{1-c})Bigr},
-where $c>0$ and the constant in the error depend only on $d$ and
-$kappa$.
+where c>0 and the constant in the error depend only on d and
+kappa.
Proof.
-Work in the configuration model with the $d$ stubs at each vertex
-ordered. Fix an ordered star type $T_0$, i.e. a distribution on
-the ordered observation $(z,x_1,...,x_d)$. Prescribe for every vertex
-its label and the labels of its $d$ ordered stubs' desired partners.
-For this fixed $T_0$ there are at most $exp(nH(T_0))$ such arrays,
-because there are $n$ vertices and the empirical distribution of their
-type is $T_0$.
-
-Now fix also $Q$, and let $m_{jk}=nd,Q_{jk}$ be the number of stubs at
-vertices of label $j$ whose desired partner has label $k$. A compatible
-matching requires $m_{jk}=m_{kj}$ for all $j,k$, and requires $m_{jj}$
-to be even for each $j$. Given these conditions, the number of uniform
-matchings of the $nd$ stubs that realize the prescription is
+Work in the configuration model with the d stubs at each vertex
+ordered. Fix an ordered star type T_0, i.e. a distribution on
+the ordered observation (z,x_1,...,x_d). Prescribe for every vertex
+its label and the labels of its d ordered stubs' desired partners.
+For this fixed T_0 there are at most exp(nH(T_0)) such arrays,
+because there are n vertices and the empirical distribution of their
+type is T_0.
+
+Now fix also Q, and let m_{jk}=nd,Q_{jk} be the number of stubs at
+vertices of label j whose desired partner has label k. A compatible
+matching requires m_{jk}=m_{kj} for all j,k, and requires m_{jj}
+to be even for each j. Given these conditions, the number of uniform
+matchings of the nd stubs that realize the prescription is
[
prod_{j<k}m_{jk}!;prod_j (m_{jj}-1)!! ,
]
-the first factor choosing the bijection between the $j$-to-$k$ stubs and
-their $k$-to-$j$ partners and the second pairing the $j$-to-$j$ stubs
-among themselves. Dividing by the total number $(nd-1)!!$ of matchings
+the first factor choosing the bijection between the j-to-k stubs and
+their k-to-j partners and the second pairing the j-to-j stubs
+among themselves. Dividing by the total number (nd-1)!! of matchings
and expanding the factorials with Stirling's formula, the logarithm of
the probability is
[
-(nd)/(2)H(Q)+O(M^2log n).
]
-(The computation: each $mlog m$ term with $m=ndQ_{jk}$ contributes
-$Q_{jk}log Q_{jk}$ times $nd$, the $-tfrac12 ndlog(nd)$ parts of the
+(The computation: each mlog m term with m=ndQ_{jk} contributes
+Q_{jk}log Q_{jk} times nd, the -tfrac12 ndlog(nd) parts of the
numerators cancel the denominator, and the residual terms are
-$O(M^2log n)$.)
+O(M^2log n).)
-Thus the expected number of labelings of ordered type $T_0$ and edge
-type $Q$ is at most $exp{nH(T_0)-tfrac{nd}2H(Q)+O(M^2log n)}$.
+Thus the expected number of labelings of ordered type T_0 and edge
+type Q is at most exp{nH(T_0)-tfrac{nd}2H(Q)+O(M^2log n)}.
Symmetrizing over neighbour orders can only increase the entropy of the
-star type, so $H(T_0)le H(T)$; it does not touch $Q$. The number of
-ordered types compatible with a given symmetrized $T$ is at most
-$exp(O(M^{d+1}log n))$, because a type is a probability
-vector on $M^{d+1}$ ordered observations. Summing over these, and using
+star type, so H(T_0)le H(T); it does not touch Q. The number of
+ordered types compatible with a given symmetrized T is at most
+exp(O(M^{d+1}log n)), because a type is a probability
+vector on M^{d+1} ordered observations. Summing over these, and using
(eq:type-count-hypothesis), changes only the sublinear error term.
Markov's inequality together with a union bound over all pairs
-$(T,Q)$ upgrades the statement about expectations to a statement
+(T,Q) upgrades the statement about expectations to a statement
holding for all pairs simultaneously with probability tending to one:
-there are only $exp(O(M^{d+1}log n))$ pairs, and each has negligible
+there are only exp(O(M^{d+1}log n)) pairs, and each has negligible
tail.
Finally, by Lemma~lem:model the probability that a uniform
@@ -825,73 +815,73 @@ passes to the uniform simple graph.
Counting alone sees only the discrete labels of a colouring. To connect
it to eigenvectors we smooth by a Gaussian field whose covariance is the
-tree covariance at height $eta$, so that the labelings of the smoothed
+tree covariance at height eta, so that the labelings of the smoothed
field approximate a genuine continuous law.
-Choose small fixed parameters $gamma,alpha>0$ with
+Choose small fixed parameters gamma,alpha>0 with
- 2ell(d+1)gamma<1,qquad
- 4ellgamma+2alpha<frac14,qquad
+ 2ell(d+1)gamma<1,
+ 4ellgamma+2alpha<frac14,
0<alpha<gamma,
-and take $alpha$ also smaller than the rigidity exponent $c_0$ of
+and take alpha also smaller than the rigidity exponent c_0 of
Proposition~prop:spectral-input. Put
eta=n^{-alpha},
and, conditionally on the graph, let
-$Y=(Y^{(1)},...,Y^{(ell)})$ be $ell$ independent Gaussian fields on
+Y=(Y^{(1)},...,Y^{(ell)}) be ell independent Gaussian fields on
the vertices, each with the covariance
Gamma=(1)/(pirho),(eta)/((A-E)^2+eta^2).
This is exactly the covariance kernel (eq:cov-definition) at
-height $eta$, so its entries approximate the tree covariance as
-$eta↓0$. For $0leεle1$ define the smoothed field
+height eta, so its entries approximate the tree covariance as
+eta↓0. For 0leεle1 define the smoothed field
[
F=ε V+Y ,
]
-where $V$ is the eigenvector field $V(v)=(u_{i(1)}(v),...,u_{i(ell)}(v))$.
-The parameter $eta$ regularizes the almost-deterministic eigenvector
-relation; the independent parameter $ε$ measures how much of the
+where V is the eigenvector field V(v)=(u_{i(1)}(v),...,u_{i(ell)}(v)).
+The parameter eta regularizes the almost-deterministic eigenvector
+relation; the independent parameter ε measures how much of the
eigenvector signal is kept. We keep the entropy inequality uniformly in
-$ε$ and later choose a small power of $n^{-1}$ when extracting
+ε and later choose a small power of n^{-1} when extracting
each fixed moment.
[Star--edge entropy]
On events of graphs of probability tending to one, uniformly over all
-choices of $V$ and all $0leεle1$,
+choices of V and all 0leεle1,
mathcal H(F_calH(F_{rm star})-frac d2mathcal H(F_calH(F_{rm edge})
ge frac1nmathcal H(Y_calH(Y_{rm global})-O(n^{-c}),
-where $Y_{rm global}$ is the full $nell$-dimensional Gaussian array.
+where Y_{rm global} is the full nell-dimensional Gaussian array.
Moreover
frac1nmathcal H(Y_calH(Y_{rm global})
=ell(frac12logfrac{2pi{rm e}eta}{pirho}-U(E))
+O(n^{-c}),
-with $U(E)=intlog|t-E|rho_d(t),dd t$ from
+with U(E)=intlog|t-E|rho_d(t),dd t from
Lemma~lem:tree-identities.
Proof.
-Step 1: quantize and count. Quantize every coordinate of $F$ into
-intervals of width $b=n^{-gamma}$, clipping all values outside
-$[-n^{gamma},n^{gamma}]$ into the two end bins. The resulting vertex
+Step 1: quantize and count. Quantize every coordinate of F into
+intervals of width b=n^{-gamma}, clipping all values outside
+[-n^{gamma},n^{gamma}] into the two end bins. The resulting vertex
alphabet has
M=(2n^{2gamma})^{ell}=n^{2ellgamma+o(1)}quadelements,
-so by $2ell(d+1)gamma<1$ the hypothesis
+so by 2ell(d+1)gamma<1 the hypothesis
(eq:type-count-hypothesis) holds with room to spare. Let
-$mathcal L$calL$ be the resulting random quantized labeling of the vertices,
-and let $T,Q$ be its star and edge histograms. Entropy of the type
-index. The total number of labelings is $exp(nH(mathcal L))$,calL))$, the
+mathcal LcalL be the resulting random quantized labeling of the vertices,
+and let T,Q be its star and edge histograms. Entropy of the type
+index. The total number of labelings is exp(nH(mathcal L)),calL)), the
number of possible type indices is at most
-$exp(O(M^{d+1}log n))$, and for a given type index the count is at
-most (eq:ent-type-count). Hence, by concavity of $H$ in its first
+exp(O(M^{d+1}log n)), and for a given type index the count is at
+most (eq:ent-type-count). Hence, by concavity of H in its first
argument,
frac1nH(mathcal L)calL)
@@ -907,31 +897,31 @@ norm of at most
C((TrGamma^2)/(n^2))^{1/2}
le frac{C}{√(neta)},
]
-because $TrGamma^2le Cn/eta$, and the smallest eigenvalue of a
-single-edge covariance is at least $ceta$ (use $|A|le d$). After
+because TrGamma^2le Cn/eta, and the smallest eigenvalue of a
+single-edge covariance is at least ceta (use |A|le d). After
whitening the two marginal blocks, the total-variation distance from the
-product law is at most $|{rm off,block}|/lambda_{min}le
-Cn^{-1/2}eta^{-3/2}$; for reference this follows by differentiating the
+product law is at most |{rm off,block}|/lambda_{min}le
+Cn^{-1/2}eta^{-3/2}; for reference this follows by differentiating the
Gaussian density along the segment joining the two covariance matrices.
Consequently, for two independent uniform oriented edges and any cell of
the quantization, the covariance of the two cell indicators is at most
-that distance. The edge histogram $Q$ is an average over the $nd$
+that distance. The edge histogram Q is an average over the nd
oriented edges, and the joint law of two distinct oriented edges is
exactly that of two independent uniform oriented edges; hence every cell
-frequency of $Q$ has variance at most
+frequency of Q has variance at most
[
frac1{nd}+C,n^{-1/2}eta^{-2}le C,n^{-1/2+2alpha},
]
-the diagonal term being smaller. There are $M^2=n^{4ellgamma+o(1)}$
-cells, so Cauchy--Schwarz bounds the mean $ell^1$ deviation
+the diagonal term being smaller. There are M^2=n^{4ellgamma+o(1)}
+cells, so Cauchy--Schwarz bounds the mean ell^1 deviation
[
EE|Q-EE Q|_1
le (M^2cdot Cn^{-1/2+2alpha})^{1/2}
=n^{2ellgamma-1/4+alpha+o(1)}=O(n^{-c}),
]
-because $4ellgamma+2alpha<tfrac14$. For two probability vectors on a
-set of size $M^2$ at $ell^1$ distance $h$, the entropies differ by at
-most $hlog(M^2)+H_{rm bin}(h)$, where $H_{rm bin}$ is the binary
+because 4ellgamma+2alpha<tfrac14. For two probability vectors on a
+set of size M^2 at ell^1 distance h, the entropies differ by at
+most hlog(M^2)+H_{rm bin}(h), where H_{rm bin} is the binary
entropy; therefore
EE H(Q)=H(EE Q)+O(n^{-c}).
@@ -939,7 +929,7 @@ entropy; therefore
Step 3: from discrete to differential entropy. Every marginal
covariance occurring here, including that of the whole array, has
-spectrum in $[ceta,C/eta]$, because $Gamma$ does and because a
+spectrum in [ceta,C/eta], because Gamma does and because a
deterministic shift does not change covariance. Unbounded binning of a
Gaussian vector and the entropy bound on a cube give
[
@@ -947,32 +937,32 @@ Gaussian vector and the entropy bound on a cube give
-elllog b-O(n^{-c});
]
the clipping error is in fact superpolynomially small, since
-$alpha<gamma$ makes the Gaussian tails beyond $n^{gamma}$
+alpha<gamma makes the Gaussian tails beyond n^{gamma}
superpolynomially unlikely and their quantized entropy contributions
superpolynomially small, uniformly in the shift. For either local
-observation, of dimension $ell(d+1)$ for the star and $2ell$ for the
+observation, of dimension ell(d+1) for the star and 2ell for the
edge, we also have the two-sided expansion
H(binned X)=mathcal H(X)-(calH(X)-(dim X)log b+O(n^{-c}).
Here is the justification, since everything downstream rests on it.
Bin averaging changes a Gaussian density, or a mixture of Gaussians, in
-total variation by at most $Cbeta^{-1/2}$, by integrating its absolute
+total variation by at most Cbeta^{-1/2}, by integrating its absolute
first derivatives along each coordinate. Clipping adds only a
superpolynomial tail error. Both resulting densities have polynomial
supremum and second-moment bounds, and such bounds convert polynomial
total variation into polynomial entropy error: subtract the common
-minimum of the two densities; the remainders have equal mass $h$ and,
-after division by $h$, entropy bounded by $O(1+log(n/h))$; mixing the
+minimum of the two densities; the remainders have equal mass h and,
+after division by h, entropy bounded by O(1+log(n/h)); mixing the
parts with a common piece costs at most the binary entropy.
-The laws $EE T$ and $EE Q$ are exactly the binned local laws.
+The laws EE T and EE Q are exactly the binned local laws.
Substituting (eq:ent-edge-concentration) and
(eq:ent-binning) into (eq:ent-quantized), the mesh terms
cancel because
-$(d+1)ell-frac d2(2ell)=ell$, and the shift $ε V$ leaves the
-differential entropy unchanged; since $F_{rm global}$ is a
-deterministic shift of $Y_{rm global}$ we obtain
+(d+1)ell-frac d2(2ell)=ell, and the shift ε V leaves the
+differential entropy unchanged; since F_{rm global} is a
+deterministic shift of Y_{rm global} we obtain
(eq:ent-star-edge).
Step 4: the global entropy. The determinant formula for a
@@ -982,88 +972,88 @@ Gaussian gives
=(ell)/(2)logfrac{2pi{rm e}eta}{pirho}
-(ell)/(n)sum_{i=1}^nlog|lambda_i-E-mathrm irmieta|,
]
-because the eigenvalues of $Gamma$ are
-$etabigl/(pirho,((lambda_i-E)^2+eta^2))$, each with
-multiplicity $ell$. The function $tmapstolog|t-E-mathrm irmieta|$ is
-Lipschitz with constant at most $eta^{-1}$. By rigidity we may replace
+because the eigenvalues of Gamma are
+etabigl/(pirho,((lambda_i-E)^2+eta^2)), each with
+multiplicity ell. The function tmapstolog|t-E-mathrm irmieta| is
+Lipschitz with constant at most eta^{-1}. By rigidity we may replace
the eigenvalues by Kesten--McKay quantiles at a total error of at most
-$ncdoteta^{-1}cdot n^{-c_0}$; the quantile sum differs from its
-integral against $rho_d$ by $O((neta)^{-1})$, by summing the
+ncdoteta^{-1}cdot n^{-c_0}; the quantile sum differs from its
+integral against rho_d by O((neta)^{-1}), by summing the
lengths of the quantile intervals; an omitted or extra bounded
-eigenvalue costs $O(log n/n)$. The choice $alpha<c_0$ makes the first
-error $o(n)$. Lemma~lem:tree-identities supplies the remaining
-regularization error $O(eta)$ in turning $log|t-E-mathrm irmieta|$ into
-$log|t-E|$. This proves (eq:ent-global-entropy).
+eigenvalue costs O(log n/n). The choice alpha<c_0 makes the first
+error o(n). Lemma~lem:tree-identities supplies the remaining
+regularization error O(eta) in turning log|t-E-mathrm irmieta| into
+log|t-E|. This proves (eq:ent-global-entropy).
The entropy inequality (eq:ent-star-edge) still contains one
nearly deterministic direction. An eigenvector satisfies
-$sum_{ysim v}u(y)=lambda u(v)$, so at a star with centre $v$ the sum
-of the $d$ neighbour values is nearly $E$ times the centre value; this
+sum_{ysim v}u(y)=lambda u(v), so at a star with centre v the sum
+of the d neighbour values is nearly E times the centre value; this
is not a random relation, and a Gaussian model fitted to the raw star
would spend a whole direction on it. We remove that direction, so that
the surviving Gaussian has identity covariance, which is what the
Hermite expansion in the tensor step needs.
-Write a star as $(Z,X)$ with $ZinR^{ell}$ (the $ell$ colour values
-at the centre) and $XinR^{d} ⊗ R^{ell}$ (the colour values at
-the $d$ neighbours). Let $P$ be the orthogonal projection of $R^d$ onto
-${bf 1}^{perp}$, the hyperplane orthogonal to the all-ones vector, and
+Write a star as (Z,X) with ZinR^{ell} (the ell colour values
+at the centre) and XinR^{d} ⊗ R^{ell} (the colour values at
+the d neighbours). Let P be the orthogonal projection of R^d onto
+{bf 1}^{perp}, the hyperplane orthogonal to the all-ones vector, and
define the linear map to the ``clean'' coordinates, and the residual,
by
[
- mathcal B(Z,X)=(Z,(PX)/(calB(Z,X)=(Z,(PX)/(sqrt a)),qquad
+ mathcal B(Z,X)=(Z,(PX)/(calB(Z,X)=(Z,(PX)/(sqrt a)),
R_{rm res}(Z,X)=sum_{i=1}^dX_i-EZ .
]
-Everything is tensored with the identity on the $ell$ colours, so
-$D:=(R ⊕ {bf 1}^{perp}) ⊗ R^{ell}$ is the space in which
-$mathcal B$calB$ takes its values. Let $w$ denote the unit vector along the
-central direction $R$ inside $R ⊕ {bf 1}^{perp}$, and put
+Everything is tensored with the identity on the ell colours, so
+D:=(R ⊕ {bf 1}^{perp}) ⊗ R^{ell} is the space in which
+mathcal BcalB takes its values. Let w denote the unit vector along the
+central direction R inside R ⊕ {bf 1}^{perp}, and put
[
- v_i=p,w+sqrt a,Pe_i,qquad
- U_i=spanop(w,v_i) ⊗ R^{ell}quad(i=1,...,d),
+ v_i=p,w+sqrt a,Pe_i,
+ U_i=spanop(w,v_i) ⊗ R^{ell}(i=1,...,d),
]
-where $e_i$ is the $i$-th standard basis vector of $R^d$. Since
-$|Pe_i|^2=(d-1)/d$ and $a(d-1)/d=1-p^2$ by (eq:a-identity), we
-get $|v_i|=1$ and $ip{w}{v_i}=p$. These two directions are the two
-observable coordinates of the $i$-th edge: the centre and the $i$-th
+where e_i is the i-th standard basis vector of R^d. Since
+|Pe_i|^2=(d-1)/d and a(d-1)/d=1-p^2 by (eq:a-identity), we
+get |v_i|=1 and ip{w}{v_i}=p. These two directions are the two
+observable coordinates of the i-th edge: the centre and the i-th
neighbour. Their Gram matrix is
-$1&pp&1$, with determinant $1-p^2>0$, so
-they are linearly independent and the coordinate map on $U_i$ is
+1&pp&1, with determinant 1-p^2>0, so
+they are linearly independent and the coordinate map on U_i is
invertible.
-Let $P_i$ be the orthogonal projection of $D$ onto $U_i$, put
-$B_V=mathcal B(V_calB(V_{rm star})$ where $V_{rm star}$ is the true
-eigenvector star, and let $G$ be an independent standard Gaussian vector
-in $D$. All entropies on $D$ and on $U_i$ are computed with their
+Let P_i be the orthogonal projection of D onto U_i, put
+B_V=mathcal B(V_calB(V_{rm star}) where V_{rm star} is the true
+eigenvector star, and let G be an independent standard Gaussian vector
+in D. All entropies on D and on U_i are computed with their
Euclidean volume.
[Projected-star entropy]
On the graph events of Lemma~ent:star-edge, uniformly for
-$0leεle1$ and for each $i$,
+0leεle1 and for each i,
mathcal H(εcalH(ε B_V+G)-frac d2mathcal H(P_i(εcalH(P_i(ε
B_V+G)) ge -O(n^{-c}).
-Moreover $B_V$ has mean zero and second-moment matrix
-$Id_D+O(n^{-1+o(1)})$, and every fixed moment of $P_iB_V$ is invariant,
-up to $O(n^{-1+o(1)})$, under the orthogonal map of $U_i$ that
-interchanges $w$ and $v_i$ and acts trivially on the colours.
+Moreover B_V has mean zero and second-moment matrix
+Id_D+O(n^{-1+o(1)}), and every fixed moment of P_iB_V is invariant,
+up to O(n^{-1+o(1)}), under the orthogonal map of U_i that
+interchanges w and v_i and acts trivially on the colours.
Proof.
The Jacobian. Describe the map
-$(Z,X)mapsto(mathcal B(Z,X),R_calB(Z,X),R_{rm res}(Z,X))$ in the orthonormal
-coordinates of $R^{d+1}=R ⊕ R{bf 1} ⊕ {bf 1}^{perp}$, one
+(Z,X)mapsto(mathcal B(Z,X),R_calB(Z,X),R_{rm res}(Z,X)) in the orthonormal
+coordinates of R^{d+1}=R ⊕ R{bf 1} ⊕ {bf 1}^{perp}, one
colour at a time. On the central line it is the identity. On the
-all-ones line, $R_{rm res}$ reads $sqrt d,t-EZ$ where $X=t{bf
-1}/sqrt d+...$, so the diagonal entry is $sqrt d$. On each of the
-$d-1$ remaining directions, $mathcal B$calB$ divides by $sqrt a$, so the
-diagonal entry is $a^{-1/2}$. The map is triangular in these
+all-ones line, R_{rm res} reads sqrt d,t-EZ where X=t{bf
+1}/sqrt d+..., so the diagonal entry is sqrt d. On each of the
+d-1 remaining directions, mathcal BcalB divides by sqrt a, so the
+diagonal entry is a^{-1/2}. The map is triangular in these
coordinates, hence its absolute determinant, per colour, is
-$sqrt d,a^{-(d-1)/2}$, and for $ell$ colours it is
-$(sqrt d,a^{-(d-1)/2})^{ell}$.
+sqrt d,a^{-(d-1)/2}, and for ell colours it is
+(sqrt d,a^{-(d-1)/2})^{ell}.
The residual is small. For the noise, note the exact identity
[
@@ -1077,9 +1067,9 @@ For the eigenvector part, the eigenvector equation gives
quadsoquad
R_{rm res}(V_{rm star})(v)=(lambda-E)V(v),
]
-and on the graph event $|lambda-E|le S^2/n$ for every selected index.
-Thus, putting $Delta=diag(lambda_{i(1)}-E,...,lambda_{i(ell)}-E)$,
-the deterministic residual of $F_{rm star}=ε V_{rm star}+Y_{rm star}$
+and on the graph event |lambda-E|le S^2/n for every selected index.
+Thus, putting Delta=diag(lambda_{i(1)}-E,...,lambda_{i(ell)}-E),
+the deterministic residual of F_{rm star}=ε V_{rm star}+Y_{rm star}
at a uniform centre has, conditionally on the location, the covariance
matrix at most
@@ -1088,7 +1078,7 @@ matrix at most
le((eta)/(pirho)+(S^4)/(n^2))Id_{ell},
because the Gaussian colours are independent and centred, so the cross
-terms vanish, and because $|Delta|le S^2/n$ entrywise. The maximum
+terms vanish, and because |Delta|le S^2/n entrywise. The maximum
entropy bound by the covariance, together with the fact that
conditioning decreases entropy, therefore gives
[
@@ -1100,36 +1090,36 @@ The relative correction inside the logarithm is
log(1+(pirho S^4)/(n^2eta))
=O(n^{-2+alpha+o(1)}),
]
-which is polynomial because $S^4=n^{o(1)}$ and $eta=n^{-alpha}$ with
-$alpha<1$.
+which is polynomial because S^4=n^{o(1)} and eta=n^{-alpha} with
+alpha<1.
The noise is standard. By the typical-star estimate of
-Proposition~prop:spectral-input at the height $E+mathrm irmieta$,
+Proposition~prop:spectral-input at the height E+mathrm irmieta,
together with the tree covariance of Lemma~lem:tree-identities,
the covariance matrix of the smoothed star noise
-$mathcal B(Y_calB(Y_{rm star})$ differs from $Id_D$ by a polynomial error
+mathcal B(Y_calB(Y_{rm star}) differs from Id_D by a polynomial error
outside a polynomially small fraction of centres. Two centred Gaussian
-vectors whose covariance matrices differ by $delta$ in operator norm
-have laws at total variation distance at most $Cdelta$ after
+vectors whose covariance matrices differ by delta in operator norm
+have laws at total variation distance at most Cdelta after
whitening; hence, using the entropy continuity argument of
Lemma~ent:star-edge, we may replace the actual noise by the
-standard Gaussian $G$ at a polynomial entropy cost. The same argument
-applies to the edge observation: its two coordinates in $U_i$ differ
-from the observations of $ε B_V+G$ along $w$ and $v_i$ by
-$O(n^{-1+o(1)})$, since the mean neighbour value is $p$ times the
+standard Gaussian G at a polynomial entropy cost. The same argument
+applies to the edge observation: its two coordinates in U_i differ
+from the observations of ε B_V+G along w and v_i by
+O(n^{-1+o(1)}), since the mean neighbour value is p times the
centre value up to the eigenvector residual. The map from the
-orthonormal coordinates on $U_i$ to the two edge coordinates has
-absolute determinant $(1-p^2)^{ell/2}$.
+orthonormal coordinates on U_i to the two edge coordinates has
+absolute determinant (1-p^2)^{ell/2}.
-Bookkeeping: the constants cancel. For a linear bijection $Phi$
-with Jacobian $J_Phi$
+Bookkeeping: the constants cancel. For a linear bijection Phi
+with Jacobian J_Phi
[
mathcal H(calH(Phi(X))=mathcal H(X)+calH(X)+log|det J_Phi| ,
]
-since the density of $Phi(X)$ is the density of $X$ divided by
-$|det J_Phi|$. Applied to the triangular map
-$Phi=(Z,X)mapsto(mathcal B(Z,X),R_calB(Z,X),R_{rm res}(Z,X))$, whose determinant
-per colour is $sqrt d,a^{-(d-1)/2}$, and combining with the
+since the density of Phi(X) is the density of X divided by
+|det J_Phi|. Applied to the triangular map
+Phi=(Z,X)mapsto(mathcal B(Z,X),R_calB(Z,X),R_{rm res}(Z,X)), whose determinant
+per colour is sqrt d,a^{-(d-1)/2}, and combining with the
conditional bound on the residual, we get
[
mathcal H(F_calH(F_{rm star})
@@ -1137,16 +1127,16 @@ conditional bound on the residual, we get
+(ell)/(2)logfrac{2pi{rm e}eta}{pirho}
-ell(frac12log d-(d-1)/(2)log a)+O(n^{-c}).
]
-The edge observation is the image of $P_iF_{rm star}$ under the
-coordinate map of $U_i$, which has determinant
-$(1-p^2)^{ell/2}$, so
+The edge observation is the image of P_iF_{rm star} under the
+coordinate map of U_i, which has determinant
+(1-p^2)^{ell/2}, so
[
mathcal H(F_calH(F_{rm edge})
=mathcal H(P_iF_calH(P_iF_{rm star})+(ell)/(2)log(1-p^2)+O(n^{-c}).
]
Substituting both formulas into (eq:ent-star-edge), and using
(eq:ent-global-entropy) for the right-hand side, the terms with
-$log(2pi{rm e}eta/pirho)$ cancel, and the remaining constant is
+log(2pi{rm e}eta/pirho) cancel, and the remaining constant is
[
-ell U(E)+ell(frac12log d-(d-1)/(2)log a)
+(dell)/(4)log(1-p^2),
@@ -1157,37 +1147,37 @@ which is exactly zero by the potential identity
mathcal H(calH(mathcal B(F_calB(F_{rm star}))
-frac d2,mathcal H(P_iF_calH(P_iF_{rm star}) ge -O(n^{-c}).
]
-Replacing the smoothed eigenvector star $F_{rm star}$ by
-$ε B_V+G$ costs polynomial entropy in exactly the same way as in
+Replacing the smoothed eigenvector star F_{rm star} by
+ε B_V+G costs polynomial entropy in exactly the same way as in
the noise-replacement step above: the covariance of
-$mathcal B(F_calB(F_{rm star})$ is $Id_D+O(n^{-1+o(1)})$ uniformly in
-$ε$, and the entropy is a continuous function of the covariance
-on the fixed finite-dimensional space $D$. This proves
+mathcal B(F_calB(F_{rm star}) is Id_D+O(n^{-1+o(1)}) uniformly in
+ε, and the entropy is a continuous function of the covariance
+on the fixed finite-dimensional space D. This proves
(eq:ent-projected-entropy).
Second moments and the reversal symmetry. Put
-$Λ=diag(lambda_{i(1)},...,lambda_{i(ell)})$. The exact
+Λ=diag(lambda_{i(1)},...,lambda_{i(ell)}). The exact
eigenvector relations, orthogonality of distinct eigenvectors, and
-uniform permutation of the $d$ neighbours give the exact second-moment
+uniform permutation of the d neighbours give the exact second-moment
identities
[
- EE ZZ^{mathsf T}sfT=EE X_iX_i^{mathsf T}sfT=Id_{ell},qquad
- EE ZX_i^{mathsf T}sfT=(Λ)/(d),qquad
+ EE ZZ^{mathsf T}sfT=EE X_iX_i^{mathsf T}sfT=Id_{ell},
+ EE ZX_i^{mathsf T}sfT=(Λ)/(d),
EE X_iX_j^{mathsf T}sfT=frac{Λ^2-d,Id_{ell}}{d(d-1)}
- quad(ine j).
+ (ine j).
]
The mean is zero because every selected eigenvector is orthogonal to the
-constant vector. Since $Λ=E,Id_ell+O(n^{-1+o(1)})$ on the graph
-event, these identities give, using $1-(E^2-d)/(d(d-1))=a$,
+constant vector. Since Λ=E,Id_ell+O(n^{-1+o(1)}) on the graph
+event, these identities give, using 1-(E^2-d)/(d(d-1))=a,
[
EE B_VB_V^{mathsf T}sfT=Id_D+O(n^{-1+o(1)}).
]
Finally, a uniform oriented edge is invariant in law under reversal. Its
-two coordinate vectors differ from the observations of $B_V$ along $w$
-and $v_i$ by $O(n^{-1+o(1)})$, and since every coordinate of $V$ is
-$n^{o(1)}$ that error stays of order $n^{-1+o(1)}$ in every fixed
-moment. The coordinate map is invertible because $|p|<1$, so reversal is
-exactly the orthogonal map of $U_i$ that interchanges $w$ and $v_i$ and
+two coordinate vectors differ from the observations of B_V along w
+and v_i by O(n^{-1+o(1)}), and since every coordinate of V is
+n^{o(1)} that error stays of order n^{-1+o(1)} in every fixed
+moment. The coordinate map is invertible because |p|<1, so reversal is
+exactly the orthogonal map of U_i that interchanges w and v_i and
acts trivially on the colours. This proves the invariance assertion.
@@ -1199,18 +1189,18 @@ Two facts do the work: a formal identity expressing the entropy deficit
as a difference of relative entropies, and a strict inequality for
symmetric tensors that forbids a nonzero first Hermite coefficient.
-Write $varphi_L$ for the standard Gaussian density on a Euclidean space
-$L$, and for a random vector $Z$ with density $f$ on $L$ define the
+Write varphi_L for the standard Gaussian density on a Euclidean space
+L, and for a random vector Z with density f on L define the
relative entropy
[
Ent(Law(Z)midvarphi_L)=int_L flog(f)/(varphi_L) .
]
-Relative entropy is nonnegative, and it vanishes only for $Z$ standard
+Relative entropy is nonnegative, and it vanishes only for Z standard
Gaussian.
-Let $Z$ be a centred random vector in $D$ with identity covariance
-matrix and finite differential entropies for $Z$ and for each $P_iZ$.
+Let Z be a centred random vector in D with identity covariance
+matrix and finite differential entropies for Z and for each P_iZ.
Then
mathcal H(Z)-calH(Z)-frac12sum_{i=1}^dmathcal H(P_iZ)calH(P_iZ)
@@ -1220,32 +1210,32 @@ Then
Proof.
-For any $X$ in a Euclidean space $L$ with a density,
-$mathcal H(X)=calH(X)=mathcal H(calH(varphi_L)-Ent(Law(X)midvarphi_L)$; this is
-the definition of relative entropy together with $intvarphilogvarphi
-=-mathcal H(calH(varphi)$. Also $mathcal H(calH(varphi_L)=(dim L)/(2)log(2pi{rm e})$.
+For any X in a Euclidean space L with a density,
+mathcal H(X)=calH(X)=mathcal H(calH(varphi_L)-Ent(Law(X)midvarphi_L); this is
+the definition of relative entropy together with intvarphilogvarphi
+=-mathcal H(calH(varphi). Also mathcal H(calH(varphi_L)=(dim L)/(2)log(2pi{rm e}).
Substituting into the left side of (eq:ent-deficit-sign), the
Gaussian entropies contribute
-$(dim D)/(2)-frac12sum_i(dim U_i)/(2)$ times
-$log(2pi{rm e})$, and this vanishes because
-$dim D=dell$ while $dim U_i=2ell$ and there are $d$ terms. What
+(dim D)/(2)-frac12sum_i(dim U_i)/(2) times
+log(2pi{rm e}), and this vanishes because
+dim D=dell while dim U_i=2ell and there are d terms. What
remains is exactly the right side.
Thus (eq:ent-projected-entropy) says that the full relative
entropy is at most half the sum of the projected relative entropies,
-up to $O(n^{-c})$. The next lemma says that a distribution which is
+up to O(n^{-c}). The next lemma says that a distribution which is
symmetric under the swap of a centre and a neighbour and has variance
one cannot have most of its relative entropy invisible to all the
projections: the small deficit forces Gaussianity.
-For an integer $jge1$ use the Euclidean norm on symmetric tensors in
-$D^{ ⊗ j}$, and let $sigma_i$ be the orthogonal map of $U_i$ that
-interchanges $w$ and $v_i$ and fixes the colour space.
+For an integer jge1 use the Euclidean norm on symmetric tensors in
+D^{ ⊗ j}, and let sigma_i be the orthogonal map of U_i that
+interchanges w and v_i and fixes the colour space.
[Tensor coercivity]
-For every fixed $jge3$ there is $c_j>0$ such that every symmetric
-$j$-tensor $H$ on $D$ satisfying
+For every fixed jge3 there is c_j>0 such that every symmetric
+j-tensor H on D satisfying
[
sigma_i^{ ⊗ j}P_i^{ ⊗ j}H=P_i^{ ⊗ j}H
qquadfor all i=1,...,d
@@ -1258,35 +1248,35 @@ obeys
Proof.
A weighted frame. Work in the two-dimensional spatial plane
-$spanop(w,v_i)$, which is the same abstract plane for every $i$ since
-$|v_i|=1$ and $ip{w}{v_i}=p$. In an orthonormal basis with
-$w=(1,0)$ and $v_i=(p,√(1-p^2))$, define the orthonormal
+spanop(w,v_i), which is the same abstract plane for every i since
+|v_i|=1 and ip{w}{v_i}=p. In an orthonormal basis with
+w=(1,0) and v_i=(p,√(1-p^2)), define the orthonormal
bisecting directions
[
- e_i^+=frac{w+v_i}{√(2(1+p))},qquad
- e_i^-=frac{w-v_i}{√(2(1-p))},qquad
- s_i=(e_i^++e_i^-)/(sqrt2),qquad
+ e_i^+=frac{w+v_i}{√(2(1+p))},
+ e_i^-=frac{w-v_i}{√(2(1-p))},
+ s_i=(e_i^++e_i^-)/(sqrt2),
t_i=(e_i^+-e_i^-)/(sqrt2).
]
-The swap fixes $e_i^+$, negates $e_i^-$, and hence interchanges $s_i$
-and $t_i$. Writing $q_0=√(1-p^2)$, a direct computation gives
+The swap fixes e_i^+, negates e_i^-, and hence interchanges s_i
+and t_i. Writing q_0=√(1-p^2), a direct computation gives
[
- ip{w}{s_i}^2=(1+q_0)/(2),qquad
- ip{w}{t_i}^2=(1-q_0)/(2),qquad
- ip{Pe_i,s_i}^2=(1-q_0)/(2),qquad
+ ip{w}{s_i}^2=(1+q_0)/(2),
+ ip{w}{t_i}^2=(1-q_0)/(2),
+ ip{Pe_i,s_i}^2=(1-q_0)/(2),
ip{Pe_i,t_i}^2=(1+q_0)/(2).
]
The strict bulk condition is
[
p^2<(4(d-1))/(d^2)
- quad⇔quad
+ ⇔
q_0>(d-2)/(d),
]
-with equality exactly at the spectral edges $|E|=2√(d-1)$; this is
-the only place where the open bulk hypothesis and $dge3$ are used. In
+with equality exactly at the spectral edges |E|=2√(d-1); this is
+the only place where the open bulk hypothesis and dge3 are used. In
particular the two numbers
[
- b_s=(q_0-(d-2)/d)/(2q_0),qquad
+ b_s=(q_0-(d-2)/d)/(2q_0),
b_t=(q_0+(d-2)/d)/(2q_0)
]
are positive and sum to one. With these weights the frame identity
@@ -1294,152 +1284,152 @@ are positive and sum to one. With these weights the frame identity
sum_{i=1}^d(b_s,s_is_i^{mathsf T}sfT+b_t,t_it_i^{mathsf T}sfT)
=Id_{R ⊕ {bf 1}^{perp}}
-holds in $R ⊕ {bf 1}^{perp}$, tensored with the identity on the
+holds in R ⊕ {bf 1}^{perp}, tensored with the identity on the
colours. To verify it, expand both sides in the orthogonal decomposition
-$R ⊕ {bf 1} ⊕ {bf 1}^{perp}$: the central component of the
+R ⊕ {bf 1} ⊕ {bf 1}^{perp}: the central component of the
left side is
-$d(b_sfrac{1+q_0}2+b_tfrac{1-q_0}2)=1$; the mixed
-$R$--${bf 1}^{perp}$ block vanishes because $s_i$ and $t_i$ have
-parallel central and $Pe_i$ components with $i$-independent
-coefficients and $sum_iPe_i=0$; and the ${bf 1}^{perp}$ block is
-$cP$ with
-$c=b_s(1-q_0)/(2)+b_t(1+q_0)/(2)=1$, because
-$sum_i(Pe_i)(Pe_i)^{mathsf T}sfT=P$.
-
-Nonnegativity. Let $R^{(1)}_{i,s}$ denote the projection of the
-first tensor slot onto $s_i$, tensored with the identity on the other
-slots, and similarly for $t_i$. Inserting (eq:ent-frame) in the
-first slot of $ip{H}{H}$ gives the exact identity
+d(b_sfrac{1+q_0}2+b_tfrac{1-q_0}2)=1; the mixed
+R--{bf 1}^{perp} block vanishes because s_i and t_i have
+parallel central and Pe_i components with i-independent
+coefficients and sum_iPe_i=0; and the {bf 1}^{perp} block is
+cP with
+c=b_s(1-q_0)/(2)+b_t(1+q_0)/(2)=1, because
+sum_i(Pe_i)(Pe_i)^{mathsf T}sfT=P.
+
+Nonnegativity. Let R^{(1)}_{i,s} denote the projection of the
+first tensor slot onto s_i, tensored with the identity on the other
+slots, and similarly for t_i. Inserting (eq:ent-frame) in the
+first slot of ip{H}{H} gives the exact identity
[
|H|^2
=sum_{i=1}^d(b_sbigl|R^{(1)}_{i,s}Hbigr|^2
+b_tbigl|R^{(1)}_{i,t}Hbigr|^2).
]
-Since $s_i,t_iin U_i$, each norm decreases when the remaining slots are
-projected onto $U_i$:
+Since s_i,t_iin U_i, each norm decreases when the remaining slots are
+projected onto U_i:
[
bigl|R^{(1)}_{i,s}Hbigr|^2
gebigl|R^{(1)}_{i,s}P_i^{(2)}... P_i^{(j)}Hbigr|^2,
]
-and likewise for $t_i$. On $P_i^{ ⊗ j}H$ the swap invariance makes
-the two norms equal, because $sigma_i$ interchanges $s_i$ and $t_i$ and
-commutes with $P_i$. Since $s_i,t_i$ are orthonormal and span the
-spatial part of $U_i$,
+and likewise for t_i. On P_i^{ ⊗ j}H the swap invariance makes
+the two norms equal, because sigma_i interchanges s_i and t_i and
+commutes with P_i. Since s_i,t_i are orthonormal and span the
+spatial part of U_i,
[
bigl|R^{(1)}_{i,s}P_i^{ ⊗ j}Hbigr|^2
+bigl|R^{(1)}_{i,t}P_i^{ ⊗ j}Hbigr|^2
=bigl|P_i^{ ⊗ j}Hbigr|^2 ,
]
-so each of the two is $frac12|P_i^{ ⊗ j}H|^2$. Using
-$b_s+b_t=1$ now gives
+so each of the two is frac12|P_i^{ ⊗ j}H|^2. Using
+b_s+b_t=1 now gives
[
|H|^2 ge frac12sum_{i=1}^dbigl|P_i^{ ⊗ j}Hbigr|^2 ,
]
which is the nonnegativity half of (eq:ent-tensor).
Where equality is possible. Suppose equality holds. Then every
-inequality above is an equality, so for every $i$ and every slot
-$age2$ the discarded part vanishes:
-$R^{(1)}_{i,s}H=R^{(1)}_{i,s}P_i^{(2)}... P_i^{(j)}H$ and likewise
-for $t$. Adding the two identities over the orthonormal pair
-$s_i,t_i$, and using symmetry of $H$ to move a projection from slot one
+inequality above is an equality, so for every i and every slot
+age2 the discarded part vanishes:
+R^{(1)}_{i,s}H=R^{(1)}_{i,s}P_i^{(2)}... P_i^{(j)}H and likewise
+for t. Adding the two identities over the orthonormal pair
+s_i,t_i, and using symmetry of H to move a projection from slot one
to any other slot, gives
P_i^{(a)}H=P_i^{ ⊗ j}Hqquadfor every slot a.
-Now take $ine k$ and three distinct slots, which exist because $jge3$:
+Now take ine k and three distinct slots, which exist because jge3:
[
P_i^{(1)}P_k^{(1)}H=P_i^{(1)}P_k^{(2)}H
=P_k^{(2)}P_i^{(3)}H=P_i^{(3)}P_k^{(1)}H=P_k^{(1)}P_i^{(1)}H .
]
-Thus the commutator $[P_i,P_k]$ annihilates the first slot of $H$, for
-all pairs $ine k$.
+Thus the commutator [P_i,P_k] annihilates the first slot of H, for
+all pairs ine k.
-Let $a_i'in{bf 1}^{perp}$ be the unit vector along $Pe_i$. On
-$R ⊕ {bf 1}^{perp}$ the spatial part of this commutator is
+Let a_i'in{bf 1}^{perp} be the unit vector along Pe_i. On
+R ⊕ {bf 1}^{perp} the spatial part of this commutator is
[
ip{a_i'}{a_k'}(a_i'(a_k')^{mathsf T}sfT-a_k'(a_i')^{mathsf T}sfT),
- qquad ip{a_i'}{a_k'}=-frac1{d-1}ne0 (dge3),
+ ip{a_i'}{a_k'}=-frac1{d-1}ne0 (dge3),
]
which is an invertible rotation generator on
-$spanop(Pe_i,Pe_k)$ and vanishes on its orthogonal complement. As
-$i,k$ range, these generate all rotations of ${bf 1}^{perp}$, so
-their common kernel in $R ⊕ {bf 1}^{perp}$ is the central line
-$R w$. Hence $H$ is supported in $(w ⊗ R^{ell})^{ ⊗ j}$.
-
-But $win U_i$, so $P_i^{ ⊗ j}H=H$, and the invariance
-$sigma_i^{ ⊗ j}P_i^{ ⊗ j}H=P_i^{ ⊗ j}H$ reads
-$sigma_i^{ ⊗ j}H=H$, i.e.
-$v_i^{ ⊗ j} ⊗ h=w^{ ⊗ j} ⊗ h$ for the
-corresponding symmetric $j$-tensor $h$ on $R^ell$. Since $w$ and
-$v_i$ are linearly independent ($|p|<1$), this forces $h=0$, hence
-$H=0$.
+spanop(Pe_i,Pe_k) and vanishes on its orthogonal complement. As
+i,k range, these generate all rotations of {bf 1}^{perp}, so
+their common kernel in R ⊕ {bf 1}^{perp} is the central line
+R w. Hence H is supported in (w ⊗ R^{ell})^{ ⊗ j}.
+
+But win U_i, so P_i^{ ⊗ j}H=H, and the invariance
+sigma_i^{ ⊗ j}P_i^{ ⊗ j}H=P_i^{ ⊗ j}H reads
+sigma_i^{ ⊗ j}H=H, i.e.
+v_i^{ ⊗ j} ⊗ h=w^{ ⊗ j} ⊗ h for the
+corresponding symmetric j-tensor h on R^ell. Since w and
+v_i are linearly independent (|p|<1), this forces h=0, hence
+H=0.
The quadratic form in (eq:ent-tensor) is continuous on the
finite-dimensional linear subspace defined by the swap constraints, and
it is strictly positive on the unit sphere of that subspace. By
-compactness it is bounded below by $c_j|H|^2$ there.
+compactness it is bounded below by c_j|H|^2 there.
We now extract Gaussian moments from the three ingredients above. The
-spaces $D$, $U_i$, projections $P_i$ and swaps $sigma_i$ keep the
+spaces D, U_i, projections P_i and swaps sigma_i keep the
geometry of the projection step.
[Entropy criterion for Gaussian moments]
-Fix an integer $qge2$. Let $B_n$ be random vectors in $D$ with
-$|B_n|le K_n$ almost surely, where $K_n=n^{o(1)}$ is deterministic.
-Suppose that for some $c_*>0$
+Fix an integer qge2. Let B_n be random vectors in D with
+|B_n|le K_n almost surely, where K_n=n^{o(1)} is deterministic.
+Suppose that for some c_*>0
[
|EE B_n|+bigl|EE B_nB_n^{mathsf T}sfT-Id_Dbigr|=O(n^{-c_*}),
]
-and that for every $1le jle q$ and every $i$
+and that for every 1le jle q and every i
[
bigl|(sigma_i^{ ⊗ j}-Id),EE(P_iB_n)^{ ⊗ j}bigr|
=O(n^{-c_*}).
]
-Let $G$ be an independent standard Gaussian in $D$ and suppose in
-addition that, uniformly for $0leεle1$,
+Let G be an independent standard Gaussian in D and suppose in
+addition that, uniformly for 0leεle1,
[
mathcal H(εcalH(ε B_n+G)-frac12sum_{i=1}^d
mathcal H(P_i(εcalH(P_i(ε B_n+G)) ge -O(n^{-c_*}).
]
-Then every fixed polynomial moment of $B_n$ of degree at most $q$
-differs from its standard Gaussian value by $O(n^{-c})$ for some $c>0$.
+Then every fixed polynomial moment of B_n of degree at most q
+differs from its standard Gaussian value by O(n^{-c}) for some c>0.
The conclusion is uniform over all families satisfying the hypotheses
-with the same $c_*$, the same error constants, and the same bound
-$K_n$.
+with the same c_*, the same error constants, and the same bound
+K_n.
Proof.
-Hermite tensors. For $yin D$ define symmetric tensors
-$He_j(y)$ by the generating identity
+Hermite tensors. For yin D define symmetric tensors
+He_j(y) by the generating identity
[
exp(ip{y}{t}-(|t|^2)/(2))
=sum_{j=0}^∞frac{ip{He_j(y)}{t^{ ⊗ j}}}{j!},
- qquad tin D .
+ tin D .
]
-Set $H_j=EEHe_j(B_n)$. We show by induction on $j$ that
+Set H_j=EEHe_j(B_n). We show by induction on j that
- |H_j|=O(n^{-c_j})qquad(1le jle q),
+ |H_j|=O(n^{-c_j})(1le jle q),
which is enough because Hermite polynomials form a basis of the ordinary
-polynomials. The first- and second-moment hypotheses give $j=1,2$.
+polynomials. The first- and second-moment hypotheses give j=1,2.
-The induction step. Fix $jge3$ and assume the preceding orders
+The induction step. Fix jge3 and assume the preceding orders
proved. Choose
[
- ε=n^{-tau},qquad
- 2jtau<frac{c_*}2,qquad (j-k)tau<frac{c_k}2quad(1le k<j),
+ ε=n^{-tau},
+ 2jtau<frac{c_*}2, (j-k)tau<frac{c_k}2quad(1le k<j),
]
-and put $u=ε/√(1+ε^2)$. Dividing every entropy
-argument in the hypothesis by $√(1+ε^2)$ leaves that
+and put u=ε/√(1+ε^2). Dividing every entropy
+argument in the hypothesis by √(1+ε^2) leaves that
inequality unchanged, because the dimensions cancel; the normalized
vector is
[
Z_u=uB_n+√(1-u^2),G,
]
-whose second-moment matrix is $Id_D+u^2O(n^{-c_*})$. Feeding this into
+whose second-moment matrix is Id_D+u^2O(n^{-c_*}). Feeding this into
(eq:ent-deficit-sign), and absorbing the second-moment error into
the Gaussian entropies, gives
@@ -1448,181 +1438,181 @@ the Gaussian entropies, gives
le O(n^{-c_*}).
-The relative entropy is essentially the $j$-th Hermite moment.
-Let $f_u$ be the density ratio of $Z_u$ to $varphi_D$. The coefficient
-of $u^k$ in the Taylor expansion of $f_u$ is
+The relative entropy is essentially the j-th Hermite moment.
+Let f_u be the density ratio of Z_u to varphi_D. The coefficient
+of u^k in the Taylor expansion of f_u is
[
frac1{k!}ip{H_k}{He_k(y)} .
]
Indeed, integrating the generating exponential against the law of
-$B_n$ gives
-$EEexp(uip{B_n}{t}-u^2|t|^2/2)$, and pairing two generating
+B_n gives
+EEexp(uip{B_n}{t}-u^2|t|^2/2), and pairing two generating
exponentials under Gaussian measure gives Hermite orthogonality
together with
[
Bigl|frac{ip{H_k}{He_k}}{k!}Bigr|_{L^2(varphi_D)}^2
=(|H_k|^2)/(k!) .
]
-The Taylor remainder through degree $j$ is
-$O(u^{j+1}n^{o(1)})$ in $L^3(varphi_D)$. At a fixed source value $b_0$
-the ratio at parameter $sin[0,u]$ is
+The Taylor remainder through degree j is
+O(u^{j+1}n^{o(1)}) in L^3(varphi_D). At a fixed source value b_0
+the ratio at parameter sin[0,u] is
[
(1-s^2)^{-dim D/2}
exp((|y|^2)/(2)-(|y-sb_0|^2)/(2(1-s^2))),
]
whose required derivatives are bounded by polynomials in
-$|y|,|b_0|$ times $Cexp(Cs|b_0||y|)$; since
-$|b_0|le K_n=n^{o(1)}$ and $u=n^{-tau+o(1)}$, their
-$L^3(varphi_D)$ norms are $n^{o(1)}$, uniformly over the source
+|y|,|b_0| times Cexp(Cs|b_0||y|); since
+|b_0|le K_n=n^{o(1)} and u=n^{-tau+o(1)}, their
+L^3(varphi_D) norms are n^{o(1)}, uniformly over the source
values. Consequently
[
int_D|f_u-1|^3,ddvarphi_Dle Cu^3n^{o(1)} ,
]
-and, since $|(1+h)log(1+h)-h-h^2/2|le C|h|^3$ for $|h|letfrac12$,
-while $int(f_u-1)ddvarphi_D=0$, we get
+and, since |(1+h)log(1+h)-h-h^2/2|le C|h|^3 for |h|letfrac12,
+while int(f_u-1)ddvarphi_D=0, we get
Ent(Law(Z_u)midvarphi_D)
=frac12int_D(f_u-1)^2ddvarphi_D+O(u^3n^{o(1)}).
-Extracting the $j$-th Hermite coefficient. Orthogonality of the
+Extracting the j-th Hermite coefficient. Orthogonality of the
Hermite tensors gives, for the squared norm above,
[
int_D(f_u-1)^2ddvarphi_D
=u^{2j}(|H_j|^2)/(j!)
+sum_{kne j}u^{2k}(|H_k|^2)/(k!) .
]
-The terms with $k<j$ are $n^{-2ktau}O(n^{-2c_k})$, hence polynomially
+The terms with k<j are n^{-2ktau}O(n^{-2c_k}), hence polynomially
small. For the whole sum we have a cruder bound that suffices: the
-left side is a $chi^2$ divergence, and conditionally on $B_n=b_0$ the
+left side is a chi^2 divergence, and conditionally on B_n=b_0 the
density ratio is
-$rho_{b_0,u}(y)=(1-u^2)^{-dim D/2}
+rho_{b_0,u}(y)=(1-u^2)^{-dim D/2}
e^{< y,mu>-|mu|^2/2}
- e^{-u^2|y|^2/(2(1-u^2))}$
-with $mu=ub_0/(1-u^2)$; a Gaussian integral then gives
-$chi^2=O(u^2(dim D+|b_0|^2))$ for small $u$, whose mean is
-$O(u^2)$, because $EE|B_n|^2=dim D+o(1)$. Therefore
+ e^{-u^2|y|^2/(2(1-u^2))}
+with mu=ub_0/(1-u^2); a Gaussian integral then gives
+chi^2=O(u^2(dim D+|b_0|^2)) for small u, whose mean is
+O(u^2), because EE|B_n|^2=dim D+o(1). Therefore
Ent(Law(Z_u)midvarphi_D)
=frac{u^{2j}}{2}(|H_j|^2)/(j!)+O(n^{-2tau+o(1)}),
-and the same identity holds on $U_i$ with $H_j$ replaced by
-$P_i^{ ⊗ j}H_j$.
+and the same identity holds on U_i with H_j replaced by
+P_i^{ ⊗ j}H_j.
The quadratic-form inequality. Substitute
(eq:ent-relative-expansion) and its projected analogue into
-(eq:ent-relative-deficit), multiply by $2j!/u^{2j}$, and use
-$u^{-2j}=n^{2jtau+o(1)}$. The result is
+(eq:ent-relative-deficit), multiply by 2j!/u^{2j}, and use
+u^{-2j}=n^{2jtau+o(1)}. The result is
|H_j|^2-frac12sum_{i=1}^dbigl|P_i^{ ⊗ j}H_jbigr|^2
le n^{-c_*+2jtau+o(1)}le n^{-delta}
-for some $delta>0$, because $2jtau<c_*/2$.
+for some delta>0, because 2jtau<c_*/2.
-Returning to the kernel. Let $K$ be the common kernel, i.e. the
-closed subspace of symmetric $j$-tensors satisfying
-$sigma_i^{ ⊗ j}P_i^{ ⊗ j}H=P_i^{ ⊗ j}H$ for every $i$.
-Write $L_i=(sigma_i^{ ⊗ j}-Id)P_i^{ ⊗ j}$, so that
-$L_iH_j$ has norm $O(n^{-c_*})$ by hypothesis, and note that
-$K=∩_iker L_i$. The equality analysis of
+Returning to the kernel. Let K be the common kernel, i.e. the
+closed subspace of symmetric j-tensors satisfying
+sigma_i^{ ⊗ j}P_i^{ ⊗ j}H=P_i^{ ⊗ j}H for every i.
+Write L_i=(sigma_i^{ ⊗ j}-Id)P_i^{ ⊗ j}, so that
+L_iH_j has norm O(n^{-c_*}) by hypothesis, and note that
+K=∩_iker L_i. The equality analysis of
Lemma~ent:tensor shows precisely that no nonzero tensor in
-$K^{perp}$ is annihilated by all the $L_i$; since $D^{ ⊗ j}$ is
-finite-dimensional and the $L_i$ are fixed, there is $c=c(d,E,j)>0$
+K^{perp} is annihilated by all the L_i; since D^{ ⊗ j} is
+finite-dimensional and the L_i are fixed, there is c=c(d,E,j)>0
with
sum_{i=1}^d|L_iH|^2 ge c;dist(H,K)^2
-for every symmetric $H$. Hence $dist(H_j,K)=O(n^{-c_*})$.
-Let $H$ be the orthogonal projection of $H_j$ onto $K$ and $H'$ the
+for every symmetric H. Hence dist(H_j,K)=O(n^{-c_*}).
+Let H be the orthogonal projection of H_j onto K and H' the
complement. The quadratic form
-$Q(X)=|X|^2-frac12sum_i|P_i^{ ⊗ j}X|^2$ is bounded on
-$D^{ ⊗ j}$, so
+Q(X)=|X|^2-frac12sum_i|P_i^{ ⊗ j}X|^2 is bounded on
+D^{ ⊗ j}, so
[
|Q(H_j)-Q(H)|le C|H_j+H|,|H'|le n^{-c_*+o(1)},
]
-where $|H_j|=n^{o(1)}$ was used. By
-(eq:ent-quadratic-small), $Q(H)le n^{-delta'}$ for some fixed
-$delta'>0$, and by Lemma~ent:tensor,
-$Q(H)ge c_j|H|^2$. Therefore
+where |H_j|=n^{o(1)} was used. By
+(eq:ent-quadratic-small), Q(H)le n^{-delta'} for some fixed
+delta'>0, and by Lemma~ent:tensor,
+Q(H)ge c_j|H|^2. Therefore
[
|H_j|le|H|+|H'|=O(n^{-delta'/2})+O(n^{-c_*})
=O(n^{-c_j})
]
-with a new fixed $c_j>0$. This closes the induction.
+with a new fixed c_j>0. This closes the induction.
-Polynomial moments. Every monomial of degree at most $q$ equals
-$ip{T}{y^{ ⊗ j}}$ for some fixed symmetric tensor $T$ and some
-$jle q$, and $y^{ ⊗ j}$ differs from
-$He_j(y)$ by lower-order polynomial terms that are
+Polynomial moments. Every monomial of degree at most q equals
+ip{T}{y^{ ⊗ j}} for some fixed symmetric tensor T and some
+jle q, and y^{ ⊗ j} differs from
+He_j(y) by lower-order polynomial terms that are
already controlled. Hence
-$EEip{T}{B_n^{ ⊗ j}}=ip{T}{H_j}+o(n^{-c})$ equals its Gaussian
+EEip{T}{B_n^{ ⊗ j}}=ip{T}{H_j}+o(n^{-c}) equals its Gaussian
value in every fixed degree. The constants in the bounds above depend
-only on $d,E,j,c_*$ and the fixed error constants; the fact that the
-choice of $ε$ depends on $j$ is harmless because only finitely
-many $jle q$ occur. This proves the lemma.
+only on d,E,j,c_* and the fixed error constants; the fact that the
+choice of ε depends on j is harmless because only finitely
+many jle q occur. This proves the lemma.
Proof. [Proof of Proposition~prop:edge-moments]
Condition on a graph in the common event of Lemma~ent:type-count
-and Proposition~prop:spectral-input, and let $B_n=B_V$, the
+and Proposition~prop:spectral-input, and let B_n=B_V, the
projected star field of the projection step. Complete
-delocalization gives the deterministic bound $|B_n|le n^{o(1)}$.
+delocalization gives the deterministic bound |B_n|le n^{o(1)}.
Lemma~ent:projected-entropy supplies the entropy hypothesis, the
-first two moments $EE B_n=0$ and
-$EE B_nB_n^{mathsf T}sfT=Id_D+O(n^{-1+o(1)})$, and the moment
-symmetries $(sigma_i^{ ⊗ j}-Id)EE(P_iB_n)^{ ⊗ j}=O(n^{-1+o(1)})$
-for every fixed $j$, all uniformly over the selected eigenindices and
-in $0leεle1$. Averaging (eq:ent-projected-entropy) over
-$i$ gives the criterion's entropy hypothesis, with the harmless factor
-$d$.
-Lemma~ent:gaussian-criterion, applied with $q$ replaced by
-$max(q,2)$, now gives that every moment of $B_n$ of degree at most $q$
-differs from the corresponding standard Gaussian moment in $D$ by
-$O(n^{-c})$, uniformly over the choices.
+first two moments EE B_n=0 and
+EE B_nB_n^{mathsf T}sfT=Id_D+O(n^{-1+o(1)}), and the moment
+symmetries (sigma_i^{ ⊗ j}-Id)EE(P_iB_n)^{ ⊗ j}=O(n^{-1+o(1)})
+for every fixed j, all uniformly over the selected eigenindices and
+in 0leεle1. Averaging (eq:ent-projected-entropy) over
+i gives the criterion's entropy hypothesis, with the harmless factor
+d.
+Lemma~ent:gaussian-criterion, applied with q replaced by
+max(q,2), now gives that every moment of B_n of degree at most q
+differs from the corresponding standard Gaussian moment in D by
+O(n^{-c}), uniformly over the choices.
From the projected star to the edge values. The observations of
-the standard Gaussian $G$ along $w$ and $v_i$ form a $2ell$-vector
-with covariance $C_ell$, because $G$ has covariance $Id_D$ in the
-$dell$ dimensions of $D$ and $ip{w}{v_i}=p$. It remains to compare
+the standard Gaussian G along w and v_i form a 2ell-vector
+with covariance C_ell, because G has covariance Id_D in the
+dell dimensions of D and ip{w}{v_i}=p. It remains to compare
these coordinates with the actual edge values
-$(V(v),V(v'))inR^{ell} ⊕ R^{ell}$ of the coloured-star count. Write $B_V=(Z,frac1{sqrt a}PX)$ as in
+(V(v),V(v'))inR^{ell} ⊕ R^{ell} of the coloured-star count. Write B_V=(Z,frac1{sqrt a}PX) as in
the projection step. Then
[
ip{v_i}{B_V}
=pip{w}{B_V}+ip{Pe_i}{PX}
=pZ+X_i-frac1dsum_{j=1}^dX_j ,
]
-and $sum_jX_j=R_{rm res}+EZ$ by definition of the residual, so
+and sum_jX_j=R_{rm res}+EZ by definition of the residual, so
[
ip{v_i}{B_V}
=X_i+(p-frac Ed)Z-frac1dR_{rm res} .
]
-Here $p=E/d$ exactly, and $|R_{rm res}|=O(n^{-1+o(1)})$
-because $|lambda-E|le S^2/n$ and $|V(v)|=n^{o(1)}$. Hence the edge
-value vector equals the $(w,v_i)$-observations of $B_V$ up to
-$O(n^{-1+o(1)})$ in every coordinate, and since all coordinates are at
-most $n^{o(1)}$ in absolute value, every fixed coordinate monomial
-changes by $O(n^{-1+o(1)})$. This is the only place where the exact
-identity $p=E/d$ enters, and it is
+Here p=E/d exactly, and |R_{rm res}|=O(n^{-1+o(1)})
+because |lambda-E|le S^2/n and |V(v)|=n^{o(1)}. Hence the edge
+value vector equals the (w,v_i)-observations of B_V up to
+O(n^{-1+o(1)}) in every coordinate, and since all coordinates are at
+most n^{o(1)} in absolute value, every fixed coordinate monomial
+changes by O(n^{-1+o(1)}). This is the only place where the exact
+identity p=E/d enters, and it is
essential.
-Several edges. For $k$ conditionally independent uniform
+Several edges. For k conditionally independent uniform
oriented edges, a coordinate monomial factors over the edges, because
the edges are chosen independently and the eigenvectors are fixed once
-the graph is. The one-edge estimate, applied with $ell$ channels,
+the graph is. The one-edge estimate, applied with ell channels,
therefore gives the product estimate with an error of order
-$k,n^{-c+o(1)}$, and $k$ is fixed. This concludes the proof of
+k,n^{-c+o(1)}, and k is fixed. This concludes the proof of
Proposition~prop:edge-moments.
The section has produced Gaussian moments at one spectral
-energy $E$, for a fixed number of eigenvectors, fixed number of edges
-and fixed moment order, with a uniform error $n^{-c}$ over graph
+energy E, for a fixed number of eigenvectors, fixed number of edges
+and fixed moment order, with a uniform error n^{-c} over graph
dependent choices of eigenvectors. Nothing here is uniform in the
moment order or the number of edges, and nothing here applies to a
window of energies; both limitations are consistent with our later use
-and are the reason the proof strategy lets $nto∞$ before any of
+and are the reason the proof strategy lets nto∞ before any of
those parameters grow.
@@ -1631,9 +1621,9 @@ We now have Gaussian moments at sampled edges. Two obstacles remain.
First, all of the entropy argument concerns a graph on which we
already know the local law and rigidity; those are statements about
smoothed eigenvalue counts, and at the end we must pass to an
-unsmoothed statement at the single energy $E$. Second, moments of
+unsmoothed statement at the single energy E. Second, moments of
eigenvector entries are not yet a point process. This section solves
-both: it constructs the microscopic limit at $E$ and identifies its
+both: it constructs the microscopic limit at E and identifies its
unmarked marginal with the Kesten--McKay density in the correct
microscopic coordinates.
@@ -1645,38 +1635,38 @@ in one line:
average over edges of G_{vw}
= (1+zeta m_n(zeta))/(d) + small,
-where $m_n(zeta)=frac1nTr(A-zeta)^{-1}$. The right side is the
-Kesten--McKay (tree) formula with the actual $m_n$ inserted
-rather than the deterministic $m_d$. Such an identity, valid at every
+where m_n(zeta)=frac1nTr(A-zeta)^{-1}. The right side is the
+Kesten--McKay (tree) formula with the actual m_n inserted
+rather than the deterministic m_d. Such an identity, valid at every
fixed height, is exactly what a stochastic Airy-type argument normally
needs and normally lacks; it is available here because the switch is an
exact graph operation, not an approximation.
-Fix four distinct vertices $v,w,x,y$ such that $(v,x)$ and $(w,y)$ are
-edges and $(v,w)$, $(x,y)$ are not. The switch deletes the two
-edges and inserts $(v,w)$ and $(x,y)$. The result is again a simple
-$d$-regular graph on the same vertex set: degrees are preserved, the new
+Fix four distinct vertices v,w,x,y such that (v,x) and (w,y) are
+edges and (v,w), (x,y) are not. The switch deletes the two
+edges and inserts (v,w) and (x,y). The result is again a simple
+d-regular graph on the same vertex set: degrees are preserved, the new
edges join previously non-adjacent pairs, and the deleted edges are
distinct. Moreover the switch is invertible, the inverse switch being
the same operation applied to the new edge pair; hence the switch is an
-involution of the finite set of simple $d$-regular graphs, and a uniform
+involution of the finite set of simple d-regular graphs, and a uniform
graph pushed forward by the switch is again uniform. In particular, for
-every function $F$ of the graph alone,
+every function F of the graph alone,
EE F(A)=EE F(A+Z),
-where $Z$ is the matrix of the switch. If the switch is not defined at a
+where Z is the matrix of the switch. If the switch is not defined at a
sampled quadruple, both sides are interpreted by applying the identity
to the identity operation; the undefined event has probability bounded
away from one and its contribution is controlled by the same error
estimates as below. Exactness holds on the event that the sampled pair
really is switchable.
-The defining convenience of the switch is that $Z$ has rank two in the
+The defining convenience of the switch is that Z has rank two in the
following explicit sense. Put
[
- u=e_v-e_y,qquad v=e_w-e_x,qquad
+ u=e_v-e_y, v=e_w-e_x,
Z=uv^{mathsf T}sfT+vu^{mathsf T}sfT.
]
Expanding,
@@ -1686,18 +1676,18 @@ Expanding,
-e_ve_x^{mathsf T}sfT-e_xe_v^{mathsf T}sfT
-e_we_y^{mathsf T}sfT-e_ye_w^{mathsf T}sfT,
]
-so $Z$ deletes $(v,x)$ and $(w,y)$ and inserts $(v,w)$ and $(x,y)$,
-as required. Because $Z$ has rank at most two, the resolvent equation
+so Z deletes (v,x) and (w,y) and inserts (v,w) and (x,y),
+as required. Because Z has rank at most two, the resolvent equation
has an explicit elementary solution.
[Rank-two resolvent identity]
-Let $W=A-zeta$ with $Imzeta>0$, $G=W^{-1}$, $G'=(W+Z)^{-1}$, and
+Let W=A-zeta with Imzeta>0, G=W^{-1}, G'=(W+Z)^{-1}, and
[
- A_{uv}=v^{mathsf T}sfTGu,qquad A_{uu}=u^{mathsf T}sfTGu,qquad
- A_{vv}=v^{mathsf T}sfTGv,qquad
+ A_{uv}=v^{mathsf T}sfTGu, A_{uu}=u^{mathsf T}sfTGu,
+ A_{vv}=v^{mathsf T}sfTGv,
D=(1+A_{uv})^2-A_{uu}A_{vv}.
]
-Then $Dneq0$ and
+Then Dneq0 and
G'_{xy}
&=G_{xy}-D^{-1}Bigl[(G_{xv}-G_{xy})
@@ -1707,48 +1697,48 @@ Then $Dneq0$ and
[(1+A_{uv})(G_{yv}-G_{yy})-A_{uu}(G_{yw}-G_{yx})]
Bigr],
-and the same formula with $(x,y)$ replaced by $(v,w)$ throughout gives
-$G'_{vw}$.
+and the same formula with (x,y) replaced by (v,w) throughout gives
+G'_{vw}.
Proof.
-From $(W+uv^{mathsf T}sfT+vu^{mathsf T}sfT)G'=I$ we get
+From (W+uv^{mathsf T}sfT+vu^{mathsf T}sfT)G'=I we get
[
G'=G-Guv^{mathsf T}sfTG'-Gvu^{mathsf T}sfTG' .
]
-Set $X=v^{mathsf T}sfTG'$ and $Y=u^{mathsf T}sfTG'$, row vectors. Multiplying
-the displayed identity on the left by $v^{mathsf T}sfT$ and by
-$u^{mathsf T}sfT$ gives the linear system
+Set X=v^{mathsf T}sfTG' and Y=u^{mathsf T}sfTG', row vectors. Multiplying
+the displayed identity on the left by v^{mathsf T}sfT and by
+u^{mathsf T}sfT gives the linear system
[
1+A_{uv}&A_{vv} A_{uu}&1+A_{uv}
X Y
=v^{mathsf T}sfTG u^{mathsf T}sfTG,
]
-whose determinant is $D$. Since $Im G$ is negative definite and $Z$ is
-symmetric with zero exponential type, $W+Z$ has its spectrum in the open
-upper half-plane, so $G'$ exists; consistency of the system gives
-$Dne0$ (otherwise $W+Z$ would have a null vector with real
+whose determinant is D. Since Im G is negative definite and Z is
+symmetric with zero exponential type, W+Z has its spectrum in the open
+upper half-plane, so G' exists; consistency of the system gives
+Dne0 (otherwise W+Z would have a null vector with real
eigenvalue). Cramer's rule gives
[
- X=D^{-1}[(1+A_{uv})v^{mathsf T}sfTG-A_{vv}u^{mathsf T}sfTG],quad
+ X=D^{-1}[(1+A_{uv})v^{mathsf T}sfTG-A_{vv}u^{mathsf T}sfTG],
Y=D^{-1}[(1+A_{uv})u^{mathsf T}sfTG-A_{uu}v^{mathsf T}sfTG].
]
-Now take the $(x,y)$ entry of $G'=G-Guv^{mathsf T}sfTG'-Gvu^{mathsf T}sfTG'$:
+Now take the (x,y) entry of G'=G-Guv^{mathsf T}sfTG'-Gvu^{mathsf T}sfTG':
[
G'_{xy}=G_{xy}-(Gu)_x(e_y^{mathsf T}sfTv^{mathsf T}sfTG')
-(Gv)_x(e_y^{mathsf T}sfTu^{mathsf T}sfTG'),
]
-and $e_y^{mathsf T}sfTv^{mathsf T}sfTG'=(G'v)_y=(X)_y$,
-$e_y^{mathsf T}sfTu^{mathsf T}sfTG'=(Y)_y$, while
-$(Gu)_x=G_{xv}-G_{xy}$ and $(Gv)_x=G_{xw}-G_{xx}$ because
-$u=e_v-e_y$ and $v=e_w-e_x$. Substituting the expressions for $X,Y$
+and e_y^{mathsf T}sfTv^{mathsf T}sfTG'=(G'v)_y=(X)_y,
+e_y^{mathsf T}sfTu^{mathsf T}sfTG'=(Y)_y, while
+(Gu)_x=G_{xv}-G_{xy} and (Gv)_x=G_{xw}-G_{xx} because
+u=e_v-e_y and v=e_w-e_x. Substituting the expressions for X,Y
proves (eq:switch-entries).
-The local law already controls the trace $m_n$ and the adjacent resolvent
+The local law already controls the trace m_n and the adjacent resolvent
entry along an edge. What it does not produce is a closed equation that
-would pin down $m_n$ at a single microscopic energy: those estimates are
+would pin down m_n at a single microscopic energy: those estimates are
smoothed, so they see averages over windows and cannot resolve individual
spacings. A two-edge switch fills that gap. It is an exact symmetry of
the uniform graph ensemble, and Woodbury's formula converts the graph
@@ -1758,133 +1748,133 @@ relation.
Conditionally on the graph, sample an infinite sequence of independent
-uniform oriented edges. For fixed $kge1$, let $W_{n,k}$ be the $ntimes2k$
-matrix whose columns are the coordinate vectors of the first $k$
-starting vertices followed by their $k$ ending vertices. Define
+uniform oriented edges. For fixed kge1, let W_{n,k} be the ntimes2k
+matrix whose columns are the coordinate vectors of the first k
+starting vertices followed by their k ending vertices. Define
M_{n,k}(z)=W_{n,k}^{mathsf T}sfT(A-E-frac zn)^{-1}W_{n,k}
- =sum_ifrac{b_ib_i^{mathsf T}sfT}{x_i-z},qquad
- b_i=W_{n,k}^{mathsf T}sfTu_i,quad Im z>0 .
+ =sum_ifrac{b_ib_i^{mathsf T}sfT}{x_i-z},
+ b_i=W_{n,k}^{mathsf T}sfTu_i, Im z>0 .
-The normalization $|u_i|^2=n$ accounts for the missing factor $n^{-1}$.
-We show that, at any fixed microscopic height, the scalar $m_n$ and the
-matrix $M_{n,k}$ concentrate on their tree values $m_*$ and $K_k(m_*)$.
+The normalization |u_i|^2=n accounts for the missing factor n^{-1}.
+We show that, at any fixed microscopic height, the scalar m_n and the
+matrix M_{n,k} concentrate on their tree values m_* and K_k(m_*).
[Endpoint control]
-There is a deterministic $xi>0$, depending only on $d,E$, such that for
-every fixed $kge1$ and $D>0$ there are constants $C_{k,D}<∞$ and
-$y_{k,D}<∞$ with the following property. For every fixed $sinR$
-and every fixed $yge y_{k,D}$,
+There is a deterministic xi>0, depending only on d,E, such that for
+every fixed kge1 and D>0 there are constants C_{k,D}<∞ and
+y_{k,D}<∞ with the following property. For every fixed sinR
+and every fixed yge y_{k,D},
limsup_{nto∞}PP(
|m_n(s+mathrm iy)-m_*|rmiy)-m_*|
+|M_{n,k}(s+mathrm iy)-K_k(m_*)rmiy)-K_k(m_*)|>y^{-xi})le C_{k,D},y^{-D}.
-The constants and the threshold do not depend on $s$, and $s+mathrm iy$rmiy$
+The constants and the threshold do not depend on s, and s+mathrm iyrmiy
is fixed before the limit.
Proof.
-It suffices to treat $kge2$, since smaller matrices are compressions;
-throughout $k,s,y$ are fixed and error exponents may decrease from one
+It suffices to treat kge2, since smaller matrices are compressions;
+throughout k,s,y are fixed and error exponents may decrease from one
occurrence to the next.
-Initial estimates. At $z_0=s+mathrm iS$,rmiS$,
+Initial estimates. At z_0=s+mathrm iS,rmiS,
Proposition~prop:spectral-input and Lemma~lem:tree-identities
-give, for some $c>0$,
+give, for some c>0,
- m_n(z_0)=m_*+O(S^{-c}),quad M_{n,k}(z_0)=K_k(m_*)+O(S^{-c}),quad
+ m_n(z_0)=m_*+O(S^{-c}), M_{n,k}(z_0)=K_k(m_*)+O(S^{-c}),
Im m_n(mathrm iS^2)+rmiS^2)+|Im M_{n,k}(mathrm iS^2)rmiS^2)|=O(1).
-For the off-block entries, independent uniform vertices $v,w$ satisfy
-$EE_{v,w}|R_{vw}|^2=n^{-2}Tr R^*R=Im m_n(z_0)/S$, so Markov bounds
-them by $O(S^{-c})$. Endpoints of distinct sampled edges are independent
+For the off-block entries, independent uniform vertices v,w satisfy
+EE_{v,w}|R_{vw}|^2=n^{-2}Tr R^*R=Im m_n(z_0)/S, so Markov bounds
+them by O(S^{-c}). Endpoints of distinct sampled edges are independent
uniform vertices, and within one edge the typical-star estimate gives the
-adjacent entry $(1+Em_*)/d$. Also $#{|x_i|le S^2}=O(S^2)=n^{o(1)}$.
+adjacent entry (1+Em_*)/d. Also #{|x_i|le S^2}=O(S^2)=n^{o(1)}.
-Concentration under a trace bound. Descend from height $S$ to $y$
-by halving the imaginary part at fixed real part $s$; let $mathcal Z_n$calZ_n$
-be the $O(log S)$ dyadic arguments. For every fixed $C<∞$, outside
-an event of limiting probability $O_{k,D,C}(y^{-D})$, simultaneously for
-$zinmathcal Z_n$,calZ_n$,
+Concentration under a trace bound. Descend from height S to y
+by halving the imaginary part at fixed real part s; let mathcal Z_ncalZ_n
+be the O(log S) dyadic arguments. For every fixed C<∞, outside
+an event of limiting probability O_{k,D,C}(y^{-D}), simultaneously for
+zinmathcal Z_n,calZ_n,
|m_n(z)|le C ⇒ M_{n,k}(z)=K_k(m_n(z))+O((Im z)^{-c_2}),
-with $c_2>0$ independent of $D,k,s,C$. To see this, compare with $z_0$:
-for $|x|>S^2$ and large $S$,
-$|(x-z)^{-1}-(x-z_0)^{-1}|le CSx^{-2}le CS^{-1}Im(x-mathrm iS^2)^rmiS^2)^{-1}$,
-so the outside-band parts are $O(S^{-1})$ (positive residues make the
+with c_2>0 independent of D,k,s,C. To see this, compare with z_0:
+for |x|>S^2 and large S,
+|(x-z)^{-1}-(x-z_0)^{-1}|le CSx^{-2}le CS^{-1}Im(x-mathrm iS^2)^rmiS^2)^{-1},
+so the outside-band parts are O(S^{-1}) (positive residues make the
matrix bound immediate). Inside the band, with
-$w_i(z)=(x_i-z)^{-1}-(x_i-z_0)^{-1}$, a bounded trace gives
-$sum_{|x_i|le S^2}|w_i(z)|^2le 2Im m_n(z)/Im z+2Im m_n(z_0)/Sle C'/Im z$.
-Replacing the marks by independent $N(0,C_k)$ vectors, each entry of
-$sum(b_ib_i^{mathsf T}sfT-C_k)w_i(z)$ has $2j$-th absolute moment
-$O_{j,k,C}((Im z)^{-j})$ (any spectral index occurring once has zero
+w_i(z)=(x_i-z)^{-1}-(x_i-z_0)^{-1}, a bounded trace gives
+sum_{|x_i|le S^2}|w_i(z)|^2le 2Im m_n(z)/Im z+2Im m_n(z_0)/Sle C'/Im z.
+Replacing the marks by independent N(0,C_k) vectors, each entry of
+sum(b_ib_i^{mathsf T}sfT-C_k)w_i(z) has 2j-th absolute moment
+O_{j,k,C}((Im z)^{-j}) (any spectral index occurring once has zero
expectation; the remaining partitions are bounded by Gaussian moments and
the weight squares). Conditionally on a graph satisfying the fixed-order
moment estimates, the same holds up to polynomial error, with only
-$n^{o(1)}$ terms. Pick $0<c_2<min(c,1/4)$; Markov at threshold
-$(Im z)^{-c_2}$ bounds the failure by $(Im z)^{-j(1-2c_2)}$, and summing
-over the dyadic heights gives $C_{j,k,C}y^{-j/2}+o(1)$, so large $j$
+n^{o(1)} terms. Pick 0<c_2<min(c,1/4); Markov at threshold
+(Im z)^{-c_2} bounds the failure by (Im z)^{-j(1-2c_2)}, and summing
+over the dyadic heights gives C_{j,k,C}y^{-j/2}+o(1), so large j
proves the claim.
A switch gives the scalar equation. Apply the cross-switch that
exchanges the ending vertices of the first two sampled edges: the pairs
-$(v_1,w_1)$ and $(v_2,w_2)$ become $(v_1,w_2)$ and $(v_2,w_1)$. This is
+(v_1,w_1) and (v_2,w_2) become (v_1,w_2) and (v_2,w_1). This is
a different involution from Lemma~lem:switch-identity, but the
Woodbury computation below applies to it identically. With probability
-$1-O_k(n^{-1})$ the $2k$ endpoints are distinct and induce exactly the
+1-O_k(n^{-1}) the 2k endpoints are distinct and induce exactly the
sampled matching; on that event the switch is a measure-preserving
involution of the restricted graph/list space. The rank-two update changes the eigenvalue counting
function by a bounded amount, so
|widehatwidehat m_n(z)-m_n(z)|m_n(z)-m_n(z)|le C/Im z .
-Write the first two pairs as $(v_1,w_1),(v_2,w_2)$ and set
-$f_1=(e_{v_1}-e_{v_2})/sqrt2$, $f_2=(e_{w_1}-e_{w_2})/sqrt2$,
-$F=(f_1,f_2)$, $B=0&-2-2&0$; the switch adds
-$FBF^{mathsf T}sfT$, and the resolvent identity reads
-$(I+UB)widehatwidehat U=U$U=U$ with $U=F^{mathsf T}sfT(A-E-z/n)^{-1}F$. Assuming
-$|m_n(z)|le C$ and applying (eq:lim-concentration-implication) and
+Write the first two pairs as (v_1,w_1),(v_2,w_2) and set
+f_1=(e_{v_1}-e_{v_2})/sqrt2, f_2=(e_{w_1}-e_{w_2})/sqrt2,
+F=(f_1,f_2), B=0&-2-2&0; the switch adds
+FBF^{mathsf T}sfT, and the resolvent identity reads
+(I+UB)widehatwidehat U=UU=U with U=F^{mathsf T}sfT(A-E-z/n)^{-1}F. Assuming
+|m_n(z)|le C and applying (eq:lim-concentration-implication) and
(eq:lim-rank-trace) to both data sets,
- U=m&qq&m+O((Im z)^{-c_2}),quad
+ U=m&qq&m+O((Im z)^{-c_2}),
widehatwidehat U=m&-qU=m&-q-q&m+O((Im z)^{-c_2}),
- quad q=(1+Em)/(d),quad m=m_n(z),
+ q=(1+Em)/(d), m=m_n(z),
-the sign change in $widehatwidehat U$U$ coming from expressing the updated pairs
-in the old columns. The off-diagonal of $(I+UB)widehatwidehat U=U$U=U$ gives
+the sign change in widehatwidehat UU coming from expressing the updated pairs
+in the old columns. The off-diagonal of (I+UB)widehatwidehat U=UU=U gives
q^2-q-m^2=O((Im z)^{-c_2}).
-As a quadratic in $m$ this has simple conjugate roots in the strict bulk;
-by Lemma~lem:tree-identities the upper root is $m_*$ and the lower
-stays a fixed distance from ${Im mge0}$. Hence, in the bounded range,
+As a quadratic in m this has simple conjugate roots in the strict bulk;
+by Lemma~lem:tree-identities the upper root is m_* and the lower
+stays a fixed distance from {Im mge0}. Hence, in the bounded range,
m_n(z)=m_*+O((Im z)^{-c_2}).
Closing the descent. No trace bound is assumed at lower steps.
-Instead, for $eta/2leeta'leeta$, term by term
-$|x-s-mathrm irmieta'|ge(eta'/eta)|x-s-mathrm irmieta|$, so
+Instead, for eta/2leeta'leeta, term by term
+|x-s-mathrm irmieta'|ge(eta'/eta)|x-s-mathrm irmieta|, so
|m_n(s+mathrm irmieta')-m_n(s+mathrm irmieta)|
le(eta-eta')/(eta')Im m_n(s+mathrm irmieta)leIm m_n(s+mathrm irmieta).
Choose a fixed trace range containing the initial value and every
next-step value of (eq:lim-halving-bound) whenever the preceding
-value is within $1$ of $m_*$, enlarged once more for the switched trace
-via (eq:lim-rank-trace); take $y$ large enough that
-(eq:lim-scalar-stability) returns within $1$ of $m_*$. Induction
+value is within 1 of m_*, enlarged once more for the switched trace
+via (eq:lim-rank-trace); take y large enough that
+(eq:lim-scalar-stability) returns within 1 of m_*. Induction
down the heights proves the trace bounds and the endpoint estimates at
-every step. Absorbing fixed constants into a smaller $xi$ proves
-(eq:lim-endpoint-control); the only $s$-dependence is the harmless
-requirement $Sge C(|s|+y+1)$, which holds before the limit.
+every step. Absorbing fixed constants into a smaller xi proves
+(eq:lim-endpoint-control); the only s-dependence is the harmless
+requirement Sge C(|s|+y+1), which holds before the limit.
-For a locally finite point measure $nu=sum_jdelta_{x_j}$, write $PV$
-for the symmetric truncation $lim_{Rto∞}sum_{|x_j|le R}$.
+For a locally finite point measure nu=sum_jdelta_{x_j}, write PV
+for the symmetric truncation lim_{Rto∞}sum_{|x_j|le R}.
[Microscopic limits]
From every sequence of admissible sizes tending to infinity one can
@@ -1893,43 +1883,43 @@ extract a subsequence along which
(nu_n,m_n,(M_{n,k})_{kge1})⇒(nu,m,(M_k)_{kge1})
]
jointly, in the local vague topology for point measures and the
-compact-open topology for analytic functions. The limit $nu=sum_jdelta_{x_j}$
+compact-open topology for analytic functions. The limit nu=sum_jdelta_{x_j}
is locally finite, with multiplicities permitted, and there are
-deterministic $xi,chi>0$ such that the following hold.
+deterministic xi,chi>0 such that the following hold.
-For every fixed $0<delta<1$, almost surely for all sufficiently large $R$,
+For every fixed 0<delta<1, almost surely for all sufficiently large R,
- m(e+mathrm iu)=m_*+O(u^rmiu)=m_*+O(u^{-xi}),qquad |e|le R, R^deltale ule R^2,
+ m(e+mathrm iu)=m_*+O(u^rmiu)=m_*+O(u^{-xi}), |e|le R, R^deltale ule R^2,
and
- nu([v,w])=rho(w-v)+O(R^{1-chi}),qquad -Rle vle wle R, Rto∞ .
+ nu([v,w])=rho(w-v)+O(R^{1-chi}), -Rle vle wle R, Rto∞ .
-On a possible extension of the probability space, for every fixed $k$
-there are standard Gaussian vectors $g_jinR^{2k}$, independent over
-$j$ conditionally on $nu$, such that
+On a possible extension of the probability space, for every fixed k
+there are standard Gaussian vectors g_jinR^{2k}, independent over
+j conditionally on nu, such that
- m(z)=m_{rm r}+PVsum_jfrac1{x_j-z},qquad
+ m(z)=m_{rm r}+PVsum_jfrac1{x_j-z},
M_k(z)=K_k(m_{rm r})+PVsum_jfrac{C_k^{1/2}g_jg_j^{mathsf T}sfTC_k^{1/2}}{x_j-z}.
-These representations are consistent as $k$ grows by keeping the first
-$k$ channels, a separate mark is assigned to each occurrence of a
-repeated point, the principal values converge locally uniformly on $C_+$,
-and $Im M_k(z)$ is positive definite there, almost surely simultaneously
-for all finite $k$ and all $zinC_+$.
+These representations are consistent as k grows by keeping the first
+k channels, a separate mark is assigned to each occurrence of a
+repeated point, the principal values converge locally uniformly on C_+,
+and Im M_k(z) is positive definite there, almost surely simultaneously
+for all finite k and all zinC_+.
Proof.
-Tightness. For fixed $L,y>0$, positivity gives
-$nu_n([-L,L])le(L^2+y^2)y^{-1}Im m_n(mathrm iy)$;rmiy)$; using
-Proposition~prop:endpoint-control with large fixed $y$ (chosen per
+Tightness. For fixed L,y>0, positivity gives
+nu_n([-L,L])le(L^2+y^2)y^{-1}Im m_n(mathrm iy);rmiy); using
+Proposition~prop:endpoint-control with large fixed y (chosen per
tolerance) gives tightness of the local counts. For compact-open
-tightness, fix a compact $mathcal KcalKsubsetC_+$ and $y>0$. For a resolvent
-sum $f$ with nonnegative residues, comparing kernels on the real line
-gives $Im f(z)le C_{mathcal K,y}calK,yIm f(mathrm iy)$rmiy)$ and
-$|f(z)-f(mathrm iy)|rmiy)|le C_{mathcal K,y}calK,yIm f(mathrm iy)$rmiy)$ for
-$zinmathcal K$;calK$; applying these to $m_n$ and, by polarization, to the
-real quadratic forms of $M_{n,k}$ bounds each fixed compact, and Cauchy's
+tightness, fix a compact mathcal KcalKsubsetC_+ and y>0. For a resolvent
+sum f with nonnegative residues, comparing kernels on the real line
+gives Im f(z)le C_{mathcal K,y}calK,yIm f(mathrm iy)rmiy) and
+|f(z)-f(mathrm iy)|rmiy)|le C_{mathcal K,y}calK,yIm f(mathrm iy)rmiy) for
+zinmathcal K;calK; applying these to m_n and, by polarization, to the
+real quadratic forms of M_{n,k} bounds each fixed compact, and Cauchy's
estimates give equicontinuity on smaller compacts. Extracting diagonally
over growing compacts and channel dimensions gives the joint limit;
local vague limits with locally bounded counts are point measures with
@@ -1937,33 +1927,33 @@ integer multiplicities, and the analytic limits have nonnegative
imaginary parts.
Conditional Gaussian marks on compact intervals. Choose nested
-compact position intervals exhausting $R$ with deterministic boundaries
-almost surely uncharged by $nu$ (possible since only countably many
+compact position intervals exhausting R with deterministic boundaries
+almost surely uncharged by nu (possible since only countably many
points have positive probability of being atoms). On each interval
truncate the point count at a fixed integer, order the prelimit indices
using graph data alone, and extract positions and marks. By
Proposition~prop:edge-moments every fixed joint moment of the marks
-is polynomially close to that of independent $N(0,C_k)$ vectors, uniformly
+is polynomially close to that of independent N(0,C_k) vectors, uniformly
over the selected indices. Diagonal selection of the graph exceptional
-events (choose $J_nto∞$ slowly enough that all orders at most $J_n$
-match to $J_n^{-1}$ on a common event of probability $ge1-J_n^{-1}$; only
+events (choose J_nto∞ slowly enough that all orders at most J_n
+match to J_n^{-1} on a common event of probability ge1-J_n^{-1}; only
finitely many fixed-order assertions enter at each stage) upgrades
fixed-moment matching to conditional weak convergence to the product
Gaussian law, uniformly over bounded lists: a list failing to converge
would admit a deterministic sequence of graphs and index lists failing
while all fixed moments converged, contradicting tightness and Gaussian
moment determinacy. Passing to the truncation limit identifies the marking
-law given $nu$, with covariance $C_k$ over the $2k$ endpoints; on an
+law given nu, with covariance C_k over the 2k endpoints; on an
extension of the probability space one assigns an independent mark
-$b_j=C_k^{1/2}g_j$ to each occurrence of a repeated point.
+b_j=C_k^{1/2}g_j to each occurrence of a repeated point.
-Removing imaginary tails. For fixed $zinC_+$ and large $L$, the
-scalar kernels satisfy, for $|x|>L$,
-$Im(x-z)^{-1}+|(x-z)^{-1}-(x-mathrm i)^rmi)^{-1}|le C_zL^{-1}Im(x-mathrm iL)^rmiL)^{-1}$;
+Removing imaginary tails. For fixed zinC_+ and large L, the
+scalar kernels satisfy, for |x|>L,
+Im(x-z)^{-1}+|(x-z)^{-1}-(x-mathrm i)^rmi)^{-1}|le C_zL^{-1}Im(x-mathrm iL)^rmiL)^{-1};
for matrix sums use positivity of the residues and bound the norm by the
trace of the positive imaginary part. By Proposition~prop:endpoint-control
-these traces at $mathrm iL$rmiL$ are tight as $Lto∞$ after the upper
-limit in $n$; the factor $L^{-1}$ removes the tails in probability, and
+these traces at mathrm iLrmiL are tight as Lto∞ after the upper
+limit in n; the factor L^{-1} removes the tails in probability, and
compact marked sums converge, so
@@ -1979,63 +1969,63 @@ are not yet excluded and are fixed below.
Uniform bounds after the limit. Pass
Proposition~prop:endpoint-control to the limit by the portmanteau
inequality on the open event of large error (its constants are independent
-of $s$), and slightly decrease $xi<1$. Fix $0<delta<1$ and take a unit
-grid on the padded region $|e|le2R$, $tfrac12R^deltale ule2R^2$,
-giving $O(R^3)$ arguments. For fixed dyadic $R$ this is a finite
+of s), and slightly decrease xi<1. Fix 0<delta<1 and take a unit
+grid on the padded region |e|le2R, tfrac12R^deltale ule2R^2,
+giving O(R^3) arguments. For fixed dyadic R this is a finite
collection in the exact limit; the failure probability is at most
-$CR^{3-delta D}$, and Borel--Cantelli over dyadic $R$ with large $D$
-controls it. The kernel identities give $|m'(z)|leIm m(z)/Im z$, and
+CR^{3-delta D}, and Borel--Cantelli over dyadic R with large D
+controls it. The kernel identities give |m'(z)|leIm m(z)/Im z, and
imaginary parts at comparable height and bounded distance are comparable,
-so filling the grid costs $O(1/u)$; padding reaches all large $R$. A
-countable set of $delta$'s gives every fixed-$delta$ conclusion. The
-limit in $n$ preceded this growing-grid argument.
+so filling the grid costs O(1/u); padding reaches all large R. A
+countable set of delta's gives every fixed-delta conclusion. The
+limit in n preceded this growing-grid argument.
From smoothing to counts. Equation~(eq:mesoscopic-limit) gives
-$nu([-R,R])=O(R)$. Integrating $pi^{-1}Im m(e+mathrm irmisqrt R)$ over
-$ein[v,w]$ yields $rho(w-v)+O(R^{1-xi/2})$. Compare with the sharp
-count using $P_{v,w}(x)=pi^{-1}int_v^w(sqrt R)/((x-e)^2+R)dd e$:
-points within $R^{3/4}$ of an endpoint number $O(R^{3/4})$; on
-$[-2R,2R]$ off the two strips, $|P_{v,w}(x)-mathbf 1_bf1_{[v,w]}(x)|=O(R^{-1/4})$,
-for total error $O(R^{3/4})$; for $|x|>2R$ the bound
-$P_{v,w}le CR^{3/2}/x^2$ with the linear count contributes
-$O(sqrt R)$. This gives (eq:discrepancy) for any
-$0<chi<min(1/4,xi/2)$.
+nu([-R,R])=O(R). Integrating pi^{-1}Im m(e+mathrm irmisqrt R) over
+ein[v,w] yields rho(w-v)+O(R^{1-xi/2}). Compare with the sharp
+count using P_{v,w}(x)=pi^{-1}int_v^w(sqrt R)/((x-e)^2+R)dd e:
+points within R^{3/4} of an endpoint number O(R^{3/4}); on
+[-2R,2R] off the two strips, |P_{v,w}(x)-mathbf 1_bf1_{[v,w]}(x)|=O(R^{-1/4}),
+for total error O(R^{3/4}); for |x|>2R the bound
+P_{v,w}le CR^{3/2}/x^2 with the linear count contributes
+O(sqrt R). This gives (eq:discrepancy) for any
+0<chi<min(1/4,xi/2).
Scalar principal values. Partial summation in (eq:discrepancy)
-gives $PVsum_{|x_j|>R}x_j^{-1}=O(R^{-chi})$ and
-$sum_{|x_j|>R}x_j^{-2}=O(R^{-1})$ (the leading density terms cancel in
+gives PVsum_{|x_j|>R}x_j^{-1}=O(R^{-chi}) and
+sum_{|x_j|>R}x_j^{-2}=O(R^{-1}) (the leading density terms cancel in
the symmetric sum on the two half-lines). Hence
-$S_nu(z)=PVsum_j(x_j-z)^{-1}$ is analytic and locally uniformly
+S_nu(z)=PVsum_j(x_j-z)^{-1} is analytic and locally uniformly
convergent, and comparing with the constant density gives
-$S_nu(mathrm iR)=rmiR)=mathrm irmipirho+o(1)$. By
-(eq:lim-difference-identities), $m-S_nu$ is constant with zero
-imaginary part; the two displays identify it as $m_{rm r}$.
+S_nu(mathrm iR)=rmiR)=mathrm irmipirho+o(1). By
+(eq:lim-difference-identities), m-S_nu is constant with zero
+imaginary part; the two displays identify it as m_{rm r}.
-Marked principal values. Fix $k$, condition on $nu$, and set
-$Z_j=b_jb_j^{mathsf T}sfT-C_k$, independent centered with bounded second
-moments. Ordered by increasing $|x_j|$, $sum_jZ_j/(x_j-mathrm i)$rmi)$
+Marked principal values. Fix k, condition on nu, and set
+Z_j=b_jb_j^{mathsf T}sfT-C_k, independent centered with bounded second
+moments. Ordered by increasing |x_j|, sum_jZ_j/(x_j-mathrm i)rmi)
converges almost surely entrywise, and
-$sum_{|x_j|>1}|Z_j|x_j^{-2}<∞$ conditionally almost surely, so
-the difference series between $z$ and $mathrm i$rmi$ converges absolutely
+sum_{|x_j|>1}|Z_j|x_j^{-2}<∞ conditionally almost surely, so
+the difference series between z and mathrm irmi converges absolutely
and locally uniformly; with the scalar PV this gives local uniform
-convergence of $S_{k,nu}(z)=PVsum_jb_jb_j^{mathsf T}sfT(x_j-z)^{-1}$.
-The linear count gives $sum_j|x_j-mathrm iR|^rmiR|^{-2}=O(R^{-1})$, so each
-entry of the centered marked series at $mathrm iR$rmiR$ has conditional
-second moment $O(R^{-1})$, whence
-$S_{k,nu}(mathrm iR)rmiR)tomathrm irmipirho,C_k$ in probability. Since
-$M_k-S_{k,nu}$ is almost surely constant in $z$ and
-$M_k(mathrm iR)rmiR)to K_k(m_*)$ in probability, that constant equals
-$K_k(m_*)-mathrm irmipirho,C_k=K_k(m_{rm r})$.
+convergence of S_{k,nu}(z)=PVsum_jb_jb_j^{mathsf T}sfT(x_j-z)^{-1}.
+The linear count gives sum_j|x_j-mathrm iR|^rmiR|^{-2}=O(R^{-1}), so each
+entry of the centered marked series at mathrm iRrmiR has conditional
+second moment O(R^{-1}), whence
+S_{k,nu}(mathrm iR)rmiR)tomathrm irmipirho,C_k in probability. Since
+M_k-S_{k,nu} is almost surely constant in z and
+M_k(mathrm iR)rmiR)to K_k(m_*) in probability, that constant equals
+K_k(m_*)-mathrm irmipirho,C_k=K_k(m_{rm r}).
-Finally, (eq:discrepancy) supplies infinitely many points and $C_k$
-is positive definite; $2k$ independent Gaussian marks span $R^{2k}$
-almost surely, so the positive contributions to $Im M_k(z)$ make it
-positive definite for every $zinC_+$, simultaneously over a countable
-set of $k$.
+Finally, (eq:discrepancy) supplies infinitely many points and C_k
+is positive definite; 2k independent Gaussian marks span R^{2k}
+almost surely, so the positive contributions to Im M_k(z) make it
+positive definite for every zinC_+, simultaneously over a countable
+set of k.
-We work with a microscopic subsequential law $nu$ supplied by
+We work with a microscopic subsequential law nu supplied by
Proposition~prop:microscopic-limit. Its endpoint marks are exactly
Gaussian, while genuine graph switches preserve its law. Combining the two
gives exact orthogonal switching symmetries; composing many small
@@ -2044,89 +2034,89 @@ orthogonal switches then inserts a Gaussian matrix perturbation.
[In plain words]
The microscopic-limit step leaves us with the microscopic limiting law and
its Gaussian endpoint marks. Those marks give a symmetry that a general
-law would not have: conjugating $M_k$ by an orthogonal $Q_U$ preserves
+law would not have: conjugating M_k by an orthogonal Q_U preserves
the law. Composing many small conjugations then produces an honest GOE
-perturbation: the natural graph trace $mathcal S_r(z)$calS_r(z)$ is approximated,
-in law and up to $o(1)$, by $m_{rm r} + Tr(H_r-z)^{-1}$, where $H_r$
+perturbation: the natural graph trace mathcal S_r(z)calS_r(z) is approximated,
+in law and up to o(1), by m_{rm r} + Tr(H_r-z)^{-1}, where H_r
is a fixed diagonal plus independent GOE noise.
-Fix $kge2$, write $M=M_k$, and for $Oin O(k)$ set
-$D_O=I_k&00&O$ and
-$B_O=D_OJ_kD_O^{mathsf T}sfT-J_k$, which is symmetric. For an analytic
-symmetric matrix function $M$ with positive definite imaginary part, put
+Fix kge2, write M=M_k, and for Oin O(k) set
+D_O=I_k&00&O and
+B_O=D_OJ_kD_O^{mathsf T}sfT-J_k, which is symmetric. For an analytic
+symmetric matrix function M with positive definite imaginary part, put
mathcal T_O(m,M)=(m-∂_zcalT_O(m,M)=(m-∂_zlogdet(I_{2k}+B_OM),,
D_O^{mathsf T}sfT(M^{-1}+B_O)^{-1}D_O).
-This is well defined: $M$ is invertible and
-$Im(M^{-1})=-M^{-*}(Im M)M^{-1}<0$, so $M^{-1}+B_O$ is invertible and
+This is well defined: M is invertible and
+Im(M^{-1})=-M^{-*}(Im M)M^{-1}<0, so M^{-1}+B_O is invertible and
its inverse has positive definite imaginary part. Only the logarithmic
derivative is used, so no branch of the logarithm is chosen.
[Continuous switching symmetry]
-For every fixed $kge2$ and every $Oin O(k)$, the joint law of $(m,M_k)$
-under $nu$ is invariant under $mathcal T_O$.calT_O$.
+For every fixed kge2 and every Oin O(k), the joint law of (m,M_k)
+under nu is invariant under mathcal T_O.calT_O.
Proof.
-Let $O$ be a permutation matrix. On the event, of probability
-$1-O_k(n^{-1})$, that the $2k$ sampled endpoints are distinct and induce
-exactly the sampled matching, the map $Amapsto A+WB_OW^{mathsf T}sfT$,
-$Wmapsto WD_O$ permutes the pairing between the two endpoint lists, is a
+Let O be a permutation matrix. On the event, of probability
+1-O_k(n^{-1}), that the 2k sampled endpoints are distinct and induce
+exactly the sampled matching, the map Amapsto A+WB_OW^{mathsf T}sfT,
+Wmapsto WD_O permutes the pairing between the two endpoint lists, is a
bijection of the restricted graph/list space preserving all weights, and
is inverted by the inverse permutation. The determinant and inverse
identities for finite-rank perturbations give
-$widehatwidehat m_n=m_n-∂_zm_n=m_n-∂_zlogdet(I_{2k}+B_OM_{n,k})$ and
-$widehatwidehat M_M_{n,k}=D_O^{mathsf T}sfT(M_{n,k}^{-1}+B_O)^{-1}D_O$ (the factor
-$n^{-1}$ in $m_n=n^{-1}Tr R$ supplies the derivative). Compact-open
+widehatwidehat m_n=m_n-∂_zm_n=m_n-∂_zlogdet(I_{2k}+B_OM_{n,k}) and
+widehatwidehat M_M_{n,k}=D_O^{mathsf T}sfT(M_{n,k}^{-1}+B_O)^{-1}D_O (the factor
+n^{-1} in m_n=n^{-1}Tr R supplies the derivative). Compact-open
convergence of analytic functions gives convergence of derivatives, and
positive-definite imaginary parts make the inverse regular, so passing to
the limit proves the claim for permutation matrices.
-For $Uin O(k)$ put $Q_U=diag(U,U)$. As $C_k$ and $K_k(m_{rm r})$
-commute with $Q_U$, the Gaussian representation
+For Uin O(k) put Q_U=diag(U,U). As C_k and K_k(m_{rm r})
+commute with Q_U, the Gaussian representation
(eq:marked-representation) shows that
-$mathcal R_U(m,M)=(m,Q_U^calR_U(m,M)=(m,Q_U^{mathsf T}sfTMQ_U)$ preserves the joint law. Since
-$Q_UD_OQ_U^{mathsf T}sfT=D_{UOU^{mathsf T}sfT}$ and
-$Q_UB_OQ_U^{mathsf T}sfT=B_{UOU^{mathsf T}sfT}$, we get
-$mathcal R_U^calR_U^{-1}mathcal T_OcalT_Omathcal R_U=calR_U=mathcal T_calT_{UOU^{mathsf T}sfT}$;
+mathcal R_U(m,M)=(m,Q_U^calR_U(m,M)=(m,Q_U^{mathsf T}sfTMQ_U) preserves the joint law. Since
+Q_UD_OQ_U^{mathsf T}sfT=D_{UOU^{mathsf T}sfT} and
+Q_UB_OQ_U^{mathsf T}sfT=B_{UOU^{mathsf T}sfT}, we get
+mathcal R_U^calR_U^{-1}mathcal T_OcalT_Omathcal R_U=calR_U=mathcal T_calT_{UOU^{mathsf T}sfT};
so the law is invariant under every orthogonal conjugate of a permutation
-switch. The maps satisfy $B_{OO'}=B_O+D_OB_{O'}D_O^{mathsf T}sfT$ and
-$D_{OO'}=D_OD_{O'}$, so $mathcal T_OcalT_Omathcal T_calT_{O'}=mathcal T_calT_{OO'}$
+switch. The maps satisfy B_{OO'}=B_O+D_OB_{O'}D_O^{mathsf T}sfT and
+D_{OO'}=D_OD_{O'}, so mathcal T_OcalT_Omathcal T_calT_{O'}=mathcal T_calT_{OO'}
(the inverse updates compose and the determinant ratios multiply before
differentiation). A transposition is a hyperplane reflection; its
orthogonal conjugates are all hyperplane reflections, and finite products
-generate $O(k)$.
+generate O(k).
-Fix a discrepancy exponent $0<chi<1/4$ from (eq:discrepancy). Use
-$r$ planar rotations through a small angle $theta$, with aggregate
-variance of order $T=rtheta^2$. A cutoff $K$ marks the finite block that
-will receive GOE noise, and a much larger temporary cutoff $L$ first turns
+Fix a discrepancy exponent 0<chi<1/4 from (eq:discrepancy). Use
+r planar rotations through a small angle theta, with aggregate
+variance of order T=rtheta^2. A cutoff K marks the finite block that
+will receive GOE noise, and a much larger temporary cutoff L first turns
the principal-value resolvent into a finite matrix. Choose once and for all
- 0<tau<(chi)/(100000),quad k=2r,quad K=r^{1/100},quad T=r^{tau},
-quad theta=√(T/r),quad L=r^{100/chi},quad rinN .
+ 0<tau<(chi)/(100000), k=2r, K=r^{1/100}, T=r^{tau},
+ theta=√(T/r), L=r^{100/chi}, rinN .
-For each $r$ let $O=O_r$ be the direct sum of $r$ planar rotations of angle
-$theta$, write $B=B_{O_r}$ and $M^0=K_k(m_{rm r})$. The real symmetric
+For each r let O=O_r be the direct sum of r planar rotations of angle
+theta, write B=B_{O_r} and M^0=K_k(m_{rm r}). The real symmetric
matrix
Gamma_r=C_k^{1/2}(B^{-1}+M^0)^{-1}C_k^{1/2}
-is well defined for large $r$: the singular values of $B$ are all
-$2|sin(theta/2)|$ and $|M^0|$ is bounded independently of $r$. Let
-$beta$ be a diagonal matrix of eigenvalues of $Gamma_r$ and set
+is well defined for large r: the singular values of B are all
+2|sin(theta/2)| and |M^0| is bounded independently of r. Let
+beta be a diagonal matrix of eigenvalues of Gamma_r and set
- s_1=Trbeta,qquad s_2=Trbeta^2 .
+ s_1=Trbeta, s_2=Trbeta^2 .
Index the limiting positions with multiplicity, put
-$I={j:|x_j|le K}$, $q_*=|I|$ and $D_I=diag(x_j:jin I)$. Conditionally
-on $nu$, let $mathcal W$calW$ be a raw GOE matrix of size $q_*$ (variances $1$
-off the diagonal and $2$ on it), independent of $nu$ apart from its
+I={j:|x_j|le K}, q_*=|I| and D_I=diag(x_j:jin I). Conditionally
+on nu, let mathcal WcalW be a raw GOE matrix of size q_* (variances 1
+off the diagonal and 2 on it), independent of nu apart from its
dimension, and set
H_r=D_I+s_1Id_{q_*}+√(s_2),mathcal WcalW .
@@ -2135,83 +2125,83 @@ dimension, and set
[Gaussian insertion]
The coefficients satisfy
- |beta|le Ctheta,qquad cTle s_2le CT,qquad |s_1|le CT
+ |beta|le Ctheta, cTle s_2le CT, |s_1|le CT
-with deterministic positive constants. As $rto∞$,
+with deterministic positive constants. As rto∞,
m_{rm r}+Tr(H_r-z)^{-1} ⇒ m(z)
-jointly at every fixed finite list of $zinC_+$, and the unscaled
-eigenvalue point measure of $H_r$ converges locally vaguely in law to
-$nu$. More precisely, generating the Gaussian marks of $M_k$
-conditionally on $nu$ separately for each $r$, and writing
-$mathcal S_r(z)=m(z)-∂_zcalS_r(z)=m(z)-∂_zlogdet(I_{2k}+BM_k(z))$, expectations of
+jointly at every fixed finite list of zinC_+, and the unscaled
+eigenvalue point measure of H_r converges locally vaguely in law to
+nu. More precisely, generating the Gaussian marks of M_k
+conditionally on nu separately for each r, and writing
+mathcal S_r(z)=m(z)-∂_zcalS_r(z)=m(z)-∂_zlogdet(I_{2k}+BM_k(z)), expectations of
every bounded smooth function with bounded derivatives of finitely many
-real and imaginary trace values differ by $o(1)$ when
-$m_{rm r}+Tr(H_r-z)^{-1}$ is replaced by $mathcal S_r(z)$.calS_r(z)$.
+real and imaginary trace values differ by o(1) when
+m_{rm r}+Tr(H_r-z)^{-1} is replaced by mathcal S_r(z).calS_r(z).
-First cut at $L$. Elementary consequences of (eq:discrepancy)
+First cut at L. Elementary consequences of (eq:discrepancy)
are
- nu([-R,R])=2rho R+O(R^{1-chi}),quad
- |PVsum_{|x_j|>R}frac1{x_j}|=O(R^{-chi}),quad
+ nu([-R,R])=2rho R+O(R^{1-chi}),
+ |PVsum_{|x_j|>R}frac1{x_j}|=O(R^{-chi}),
sum_{|x_j|>R}frac1{x_j^2}=O(R^{-1}),
the last two by partial summation on the two half-lines. Cut
[
M_L(z)=M^0+sum_{|x_j|le L}C_k^{1/2}g_jg_j^top C_k^{1/2}(x_j-z)^{-1}.
]
-For fixed $zinC_+$,
+For fixed zinC_+,
- |M-M_L|+|M'-M_L'|=O_{PP}(r^2L^{-chi}),qquad |M'|=O_{PP}(r^2),
+ |M-M_L|+|M'-M_L'|=O_{PP}(r^2L^{-chi}), |M'|=O_{PP}(r^2),
-since the mean tail is $O(L^{-chi})C_k$ (and $O(L^{-1})C_k$ for the
+since the mean tail is O(L^{-chi})C_k (and O(L^{-1})C_k for the
derivative), each centered entry of the Gaussian tail has variance
-$O(L^{-1})$, and there are $(4r)^2$ entries. Keeping only $|x_j|le r^4$
-(of which there are $sim2rho r^4$) and using Gaussian fourth moments
+O(L^{-1}), and there are (4r)^2 entries. Keeping only |x_j|le r^4
+(of which there are sim2rho r^4) and using Gaussian fourth moments
gives
- Im M(z)ge r^{-5}Id_{4r},qquad Im M_L(z)ge r^{-5}Id_{4r}
-
-with probability tending to one (using $Lgg r^4$). As $B^{-1}$ is real
-symmetric, this bounds $|(B^{-1}+M)^{-1}|$ and $|(B^{-1}+M_L)^{-1}|$ by
-$r^5$, so cutting both scalar and marked sums in
-$mathcal S_r(z)=m(z)-calS_r(z)=m(z)-Tr(B^{-1}+M(z))^{-1}M'(z)$ costs $o_{PP}(1)$: the
-marked error is $O_{PP}(rcdot r^5cdot r^{-98})+O_{PP}(rcdot r^{10}
-cdot r^{-98}cdot r^2)=O_{PP}(r^{-85})$, and the scalar tail is
-$O(L^{-chi})$. Diagonalizing $Gamma_r$ and rotating the independent
-Gaussian row vectors in channel space gives a matrix $mathcal G$calG$ of
-independent standard Gaussian entries with $4r$ columns, and the
+ Im M(z)ge r^{-5}Id_{4r}, Im M_L(z)ge r^{-5}Id_{4r}
+
+with probability tending to one (using Lgg r^4). As B^{-1} is real
+symmetric, this bounds |(B^{-1}+M)^{-1}| and |(B^{-1}+M_L)^{-1}| by
+r^5, so cutting both scalar and marked sums in
+mathcal S_r(z)=m(z)-calS_r(z)=m(z)-Tr(B^{-1}+M(z))^{-1}M'(z) costs o_{PP}(1): the
+marked error is O_{PP}(rcdot r^5cdot r^{-98})+O_{PP}(rcdot r^{10}
+cdot r^{-98}cdot r^2)=O_{PP}(r^{-85}), and the scalar tail is
+O(L^{-chi}). Diagonalizing Gamma_r and rotating the independent
+Gaussian row vectors in channel space gives a matrix mathcal GcalG of
+independent standard Gaussian entries with 4r columns, and the
finite-rank inverse identity (Woodbury) yields
- mathcal S_r(z)=m_calS_r(z)=m_{rm r}+Tr(D+A'-z)^{-1}+o_{PP}(1),qquad
- A'=mathcal GcalGbetamathcal G^calG^top,quad D=diag(x_j:|x_j|le L).
+ mathcal S_r(z)=m_calS_r(z)=m_{rm r}+Tr(D+A'-z)^{-1}+o_{PP}(1),
+ A'=mathcal GcalGbetamathcal G^calG^top, D=diag(x_j:|x_j|le L).
The coefficient bounds follow from
-$(B^{-1}+M^0)^{-1}=(Id+BM^0)^{-1}B=B+O(theta^2)$: every singular value
-is comparable to $theta$, so $|beta|=O(theta)$ and $s_2asymp
-rtheta^2=T$; and $s_1=Tr(C_kB)+O(rtheta^2)$ with
-$Tr(C_kB)=2p,Tr(O-Id_k)=O(rtheta^2)$.
+(B^{-1}+M^0)^{-1}=(Id+BM^0)^{-1}B=B+O(theta^2): every singular value
+is comparable to theta, so |beta|=O(theta) and s_2asymp
+rtheta^2=T; and s_1=Tr(C_kB)+O(rtheta^2) with
+Tr(C_kB)=2p,Tr(O-Id_k)=O(rtheta^2).
-Removing the outer positions. Let $F={j:K<|x_j|le L}$ and put
-$h=T^2$, $U_I=(|D_I|+h)^{-1/2}$, $U_F=|D_F|^{-1/2}$. Partial summation
+Removing the outer positions. Let F={j:K<|x_j|le L} and put
+h=T^2, U_I=(|D_I|+h)^{-1/2}, U_F=|D_F|^{-1/2}. Partial summation
using (eq:sw-tail-sums) gives
- Tr U_I^2+Tr U_F^2=O(log r),qquad |U_I^2|le h^{-1},quad
+ Tr U_I^2+Tr U_F^2=O(log r), |U_I^2|le h^{-1},
|U_F^2|le K^{-1}.
-In the estimates below, $polylog(r)$ denotes a fixed power
-of $log r$ whose exponent may rise from one occurrence to the next.
+In the estimates below, polylog(r) denotes a fixed power
+of log r whose exponent may rise from one occurrence to the next.
[Weighted block estimates]
-For $mathcal AcalAin{I,F}$ put $a_I=h$, $a_F=K$. With probability tending
+For mathcal AcalAin{I,F} put a_I=h, a_F=K. With probability tending
to one,
bigl|U_{mathcal A}calA(A'_{mathcal AcalAmathcal A}calA-s_1Id)U_{mathcal A}calAbigr|
lepolylog(r)(sqrt{T/a_{mathcal A}calA}+theta),
-and, for $P_0=U_IA'_{IF}U_F$,
+and, for P_0=U_IA'_{IF}U_F,
|P_0|_{HS}^2le Tpolylog(r).
@@ -2219,67 +2209,67 @@ and, for $P_0=U_IA'_{IF}U_F$,
Proof.
The centered matrix in (eq:sw-bernstein) is a sum of independent
-self-adjoint $sum_{alpha=1}^{4r}beta_{alphaalpha}(v_alpha
-v_alpha^top-P)$ with $P=U_{mathcal A}calA^2$ and $v_alpha$ Gaussian of
-covariance $P$. Gaussian fourth moments give
-$EE[(v_alpha v_alpha^top)^2]=(Tr P)P+2P^2$. Truncating each Gaussian
-column at $|entry|lelog r$ (a superpolynomially small event, with
+self-adjoint sum_{alpha=1}^{4r}beta_{alphaalpha}(v_alpha
+v_alpha^top-P) with P=U_{mathcal A}calA^2 and v_alpha Gaussian of
+covariance P. Gaussian fourth moments give
+EE[(v_alpha v_alpha^top)^2]=(Tr P)P+2P^2. Truncating each Gaussian
+column at |entry|lelog r (a superpolynomially small event, with
superpolynomially small centering correction) leaves summands of norm at
-most $C|beta|(log r)^2Tr Plethetapolylog(r)$, and
-bounds the matrix variance parameter by $Cs_2(Tr P)|P|le
-(T/a_{mathcal A}calA)polylog(r)$. The bounded matrix Bernstein
-inequality gives a two-sided tail $2|mathcal A|calA|exp(-ct^2/
-(sigma_{rm mat}^2+bt))$, and a large power of $log r$ in $t$
+most C|beta|(log r)^2Tr Plethetapolylog(r), and
+bounds the matrix variance parameter by Cs_2(Tr P)|P|le
+(T/a_{mathcal A}calA)polylog(r). The bounded matrix Bernstein
+inequality gives a two-sided tail 2|mathcal A|calA|exp(-ct^2/
+(sigma_{rm mat}^2+bt)), and a large power of log r in t
proves (eq:sw-bernstein). For the cross block, independence of
-distinct Gaussian rows gives $EE|P_0|_{HS}^2=s_2(Tr U_I^2)(Tr U_F^2)$,
+distinct Gaussian rows gives EE|P_0|_{HS}^2=s_2(Tr U_I^2)(Tr U_F^2),
so Markov with (eq:sw-weight-bounds) gives (eq:sw-cross-block).
-Fix $zinC_+$ and write $G_F=(D_F+A'_{FF}-z)^{-1}$,
-$G_I=(D_I+A'_{II}-z)^{-1}$.
+Fix zinC_+ and write G_F=(D_F+A'_{FF}-z)^{-1},
+G_I=(D_I+A'_{II}-z)^{-1}.
[Far block and Schur correction]
-Let $delta_F=polylog(r)((1+T)/K+√(T/K)+theta)$.
-Then $delta_F=o(1)$ and, with probability tending to one,
+Let delta_F=polylog(r)((1+T)/K+√(T/K)+theta).
+Then delta_F=o(1) and, with probability tending to one,
- |U_F^{-1}(G_F-D_F^{-1})U_F^{-1}|ledelta_F=o(1),qquad
+ |U_F^{-1}(G_F-D_F^{-1})U_F^{-1}|ledelta_F=o(1),
Tr G_F=o_{PP}(1),
-and, for $Σ=A'_{IF}G_FA'_{FI}$,
+and, for Σ=A'_{IF}G_FA'_{FI},
|U_IΣ U_I|=o_{PP}(T^{-5}).
Proof.
-Conjugating by $U_F$,
-$U_F(D_F+A'_{FF}-z)U_F=sign(D_F)+U_F(A'_{FF}-z)U_F$, and the
-second term has norm at most $delta_F=o(1)$ by
-Lemma~sw:weighted-blocks, $|s_1|=O(T)$ and (eq:sw-weight-bounds);
+Conjugating by U_F,
+U_F(D_F+A'_{FF}-z)U_F=sign(D_F)+U_F(A'_{FF}-z)U_F, and the
+second term has norm at most delta_F=o(1) by
+Lemma~sw:weighted-blocks, |s_1|=O(T) and (eq:sw-weight-bounds);
a Neumann expansion proves (eq:sw-far-resolvent), so
-$Q_F=U_F^{-1}G_FU_F^{-1}$ has bounded norm. Also
-$|Tr(G_F-D_F^{-1})|ledelta_FTr U_F^2=o(1)$, so $Tr G_F=o_{PP}(1)$.
-
-For $Σ$, replacing $G_F$ by $D_F^{-1}$ costs at most
-$|P_0|_{HS}^2delta_Fle Tpolylog(r)delta_F$ in the
-$U_I$-weighted norm. For $A'_{IF}D_F^{-1}A'_{FI}$, condition on the inner
-block and set $Q=mathcal G_IcalG_Ibeta^2mathcal G_I^calG_I^top$; the columns $a_j$
-of $A'_{IF}$, $jin F$, are independent centered Gaussian with covariance
-$Q$, so $EE[sum_{jin F}a_ja_j^top/x_jmidcdot]=Qsum_{jin F}x_j^{-1}$
-and the entrywise variance is $(Q_{aa}Q_{bb}+Q_{ab}^2)sum_{jin F}x_j^{-2}$.
-With $EE[Q_{aa}^2]=s_2^2+2Trbeta^4=O(T^2)$ and $|Q_{ab}|^2le Q_{aa}Q_{bb}$,
+Q_F=U_F^{-1}G_FU_F^{-1} has bounded norm. Also
+|Tr(G_F-D_F^{-1})|ledelta_FTr U_F^2=o(1), so Tr G_F=o_{PP}(1).
+
+For Σ, replacing G_F by D_F^{-1} costs at most
+|P_0|_{HS}^2delta_Fle Tpolylog(r)delta_F in the
+U_I-weighted norm. For A'_{IF}D_F^{-1}A'_{FI}, condition on the inner
+block and set Q=mathcal G_IcalG_Ibeta^2mathcal G_I^calG_I^top; the columns a_j
+of A'_{IF}, jin F, are independent centered Gaussian with covariance
+Q, so EE[sum_{jin F}a_ja_j^top/x_jmidcdot]=Qsum_{jin F}x_j^{-1}
+and the entrywise variance is (Q_{aa}Q_{bb}+Q_{ab}^2)sum_{jin F}x_j^{-2}.
+With EE[Q_{aa}^2]=s_2^2+2Trbeta^4=O(T^2) and |Q_{ab}|^2le Q_{aa}Q_{bb},
(eq:sw-tail-sums) and (eq:sw-weight-bounds) give
-$EE|U_IA'_{IF}D_F^{-1}A'_{FI}U_I|_{HS}^2le C T^2(log r)^2
-(K^{-2chi}+K^{-1})$, so by Markov this norm is
-$Tpolylog(r)(K^{-chi}+K^{-1/2})$ with probability tending to
-one. Combining and multiplying by $T^5$, the exponents
-$6tau-chi/100$, $tfrac{13}2tau-tfrac1{200}$ and
-$tfrac{13}2tau-tfrac12$ are all negative under (eq:sw-parameters),
+EE|U_IA'_{IF}D_F^{-1}A'_{FI}U_I|_{HS}^2le C T^2(log r)^2
+(K^{-2chi}+K^{-1}), so by Markov this norm is
+Tpolylog(r)(K^{-chi}+K^{-1/2}) with probability tending to
+one. Combining and multiplying by T^5, the exponents
+6tau-chi/100, tfrac{13}2tau-tfrac1{200} and
+tfrac{13}2tau-tfrac12 are all negative under (eq:sw-parameters),
giving (eq:sw-schur-small).
[Inner resolvent and trace cut]
-For every fixed $zinC_+$,
+For every fixed zinC_+,
|U_I^{-1}G_I(z)U_I^{-1}|=O_{PP}(h),
@@ -2287,85 +2277,85 @@ and
Tr(D+A'-z)^{-1}=Tr(D_I+A'_{II}-z)^{-1}+o_{PP}(1),
-jointly for any fixed finite list of $z$'s.
+jointly for any fixed finite list of z's.
Proof.
-Every diagonal entry of $U_I(D_I-mathrm ih)U_I$rmih)U_I$ has modulus at least
-$1/sqrt2$, and the added term has norm $o(1)$ by
-Lemma~sw:weighted-blocks, $|s_1|=O(T)$ and (eq:sw-weight-bounds).
-Hence $|U_I^{-1}G_I(mathrm ih)U_I^rmih)U_I^{-1}|le C$, and
-$G_I(mathrm ih)^*G_I(rmih)^*G_I(mathrm ih)=h^rmih)=h^{-1}Im G_I(mathrm ih)$rmih)$ gives
-$|G_I(mathrm ih)U_I^rmih)U_I^{-1}|le Ch^{-1/2}$. The twice-iterated resolvent
-identity $G_I(z)=G_I(mathrm ih)+(z-rmih)+(z-mathrm ih)G_I(rmih)G_I(mathrm ih)^2+rmih)^2+
-(z-mathrm ih)^2G_I(rmih)^2G_I(mathrm ih)G_I(z)G_I(rmih)G_I(z)G_I(mathrm ih)$,rmih)$, conjugated by
-$U_I^{-1}$, bounds the three terms by $C$, $C|z-mathrm ih|/h$rmih|/h$ and
-$C|z-mathrm ih|^2/(hrmih|^2/(hIm z)$, proving (eq:sw-inner-resolvent).
+Every diagonal entry of U_I(D_I-mathrm ih)U_Irmih)U_I has modulus at least
+1/sqrt2, and the added term has norm o(1) by
+Lemma~sw:weighted-blocks, |s_1|=O(T) and (eq:sw-weight-bounds).
+Hence |U_I^{-1}G_I(mathrm ih)U_I^rmih)U_I^{-1}|le C, and
+G_I(mathrm ih)^*G_I(rmih)^*G_I(mathrm ih)=h^rmih)=h^{-1}Im G_I(mathrm ih)rmih) gives
+|G_I(mathrm ih)U_I^rmih)U_I^{-1}|le Ch^{-1/2}. The twice-iterated resolvent
+identity G_I(z)=G_I(mathrm ih)+(z-rmih)+(z-mathrm ih)G_I(rmih)G_I(mathrm ih)^2+rmih)^2+
+(z-mathrm ih)^2G_I(rmih)^2G_I(mathrm ih)G_I(z)G_I(rmih)G_I(z)G_I(mathrm ih),rmih), conjugated by
+U_I^{-1}, bounds the three terms by C, C|z-mathrm ih|/hrmih|/h and
+C|z-mathrm ih|^2/(hrmih|^2/(hIm z), proving (eq:sw-inner-resolvent).
For (eq:sw-second-cut), the upper-left block of the full inverse is
-$widetilde G_I=(G_I^{-1}-Σ)^{-1}$. With $R_I=U_I^{-1}G_IU_I^{-1}$ and
-$S_I=U_IΣ U_I$, a Neumann expansion using
+widetilde G_I=(G_I^{-1}-Σ)^{-1}. With R_I=U_I^{-1}G_IU_I^{-1} and
+S_I=U_IΣ U_I, a Neumann expansion using
(eq:sw-inner-resolvent) and (eq:sw-schur-small) gives
-$|U_I^{-1}(widetilde G_I-G_I)U_I^{-1}|le O(h^2)o(T^{-5})=o(T^{-1})$, so
-$Trwidetilde G_I-Tr G_I=o_{PP}(1)$. The full block-inverse trace is
-$Trwidetilde G_I+Tr G_F+Tr(widetilde G_IA'_{IF}G_F^2A'_{FI})$; the
-middle term is $o_{PP}(1)$, and with
-$Q_FU_F^2Q_F=U_F^{-1}G_F^2U_F^{-1}$ of norm $O(K^{-1})$ the last term is at
-most $|U_I^{-1}widetilde G_IU_I^{-1}||P_0|_{HS}^2|Q_FU_F^2Q_F|
-=polylog(r)O(T^3/K)=o_{PP}(1)$.
+|U_I^{-1}(widetilde G_I-G_I)U_I^{-1}|le O(h^2)o(T^{-5})=o(T^{-1}), so
+Trwidetilde G_I-Tr G_I=o_{PP}(1). The full block-inverse trace is
+Trwidetilde G_I+Tr G_F+Tr(widetilde G_IA'_{IF}G_F^2A'_{FI}); the
+middle term is o_{PP}(1), and with
+Q_FU_F^2Q_F=U_F^{-1}G_F^2U_F^{-1} of norm O(K^{-1}) the last term is at
+most |U_I^{-1}widetilde G_IU_I^{-1}||P_0|_{HS}^2|Q_FU_F^2Q_F|
+=polylog(r)O(T^3/K)=o_{PP}(1).
[Gaussian replacement]
-Conditional on almost every $nu$, bounded smooth tests with bounded
+Conditional on almost every nu, bounded smooth tests with bounded
derivatives of finitely many trace values have expectations differing by
-$o(1)$ between $D_I+A'_{II}$ and
-$H_r=D_I+s_1Id_{q_*}+√(s_2),mathcal W$.calW$.
+o(1) between D_I+A'_{II} and
+H_r=D_I+s_1Id_{q_*}+√(s_2),mathcal W.calW.
Proof.
-Write $A'_{II}-s_1Id_{q_*}=sum_{alpha=1}^{4r}beta_{alphaalpha}
-(g_alpha g_alpha^top-Id_{q_*})$ with independent standard Gaussian
-$g_alpha$. Since $Cov(g_ag_b,g_cg_d)=delta_{ac}delta_{bd}+delta_{ad}
-delta_{bc}$, each $g_alpha g_alpha^top-Id$ and a raw GOE matrix have
+Write A'_{II}-s_1Id_{q_*}=sum_{alpha=1}^{4r}beta_{alphaalpha}
+(g_alpha g_alpha^top-Id_{q_*}) with independent standard Gaussian
+g_alpha. Since Cov(g_ag_b,g_cg_d)=delta_{ac}delta_{bd}+delta_{ad}
+delta_{bc}, each g_alpha g_alpha^top-Id and a raw GOE matrix have
equal mean (zero) and entry covariances; replace the summands one at a time
-by $beta_{alphaalpha}mathcal W_calW_alpha$ with independent raw GOE
-$mathcal W_calW_alpha$ (negative coefficients allowed, both covariances scaling
-by $beta_{alphaalpha}^2$). For either kind of summand $X$,
-$EE|X|^3le Ctheta^3q_*^3$ (Gaussian moments of $|g|^2$; Hilbert--
+by beta_{alphaalpha}mathcal W_calW_alpha with independent raw GOE
+mathcal W_calW_alpha (negative coefficients allowed, both covariances scaling
+by beta_{alphaalpha}^2). For either kind of summand X,
+EE|X|^3le Ctheta^3q_*^3 (Gaussian moments of |g|^2; Hilbert--
Schmidt norm for GOE). Differentiating the resolvent in a symmetric
-direction $Delta$ gives trace derivatives bounded by $C_{z,j}q_*|Delta|^j$,
-so for bounded smooth $Phi$ the third directional derivative is at most
-$C_{z,Phi}q_*^3|Delta|^3$ at every real symmetric intermediate matrix.
+direction Delta gives trace derivatives bounded by C_{z,j}q_*|Delta|^j,
+so for bounded smooth Phi the third directional derivative is at most
+C_{z,Phi}q_*^3|Delta|^3 at every real symmetric intermediate matrix.
Taylor expansion to second order cancels means and covariances in each
replacement, and the summed third-order remainder is
-$C_{z,Phi}q_*^6rtheta^3=O(r^{-0.44+(3/2)tau})=o(1)$. Finally
-$sum_alphabeta_{alphaalpha}mathcal W_calW_alpha$ has the law
-$√(s_2),mathcal W$.calW$.
+C_{z,Phi}q_*^6rtheta^3=O(r^{-0.44+(3/2)tau})=o(1). Finally
+sum_alphabeta_{alphaalpha}mathcal W_calW_alpha has the law
+√(s_2),mathcal W.calW.
Proof. [Completion of Proposition~prop:gaussian-insertion]
The coefficient bounds were proved after (eq:sw-finite-matrix). That
equation, Lemma~sw:trace-cut and Lemma~sw:lindeberg prove the
-conditional comparison with $mathcal S_r$.calS_r$. Bounded convergence averages
+conditional comparison with mathcal S_r.calS_r. Bounded convergence averages
the conditional error, and by Proposition~prop:continuous-switching
-the unconditional law of $mathcal S_r$calS_r$ is the law of $m$ for every $r$;
+the unconditional law of mathcal S_rcalS_r is the law of m for every r;
this proves (eq:sw-trace-limit).
-For the point measures, let $nu^{(r)}$ be the eigenvalue counting measure
-of $H_r$ and $sigma_r(dd x)=nu^{(r)}(dd x)/(1+x^2)$; its total mass
-$ImTr(H_r-mathrm i)^rmi)^{-1}$ is tight by (eq:sw-trace-limit), so
-$sigma_r$ is tight on the one-point compactification $overlineR$. For
-$z=e+mathrm irmieta$ the kernel $k_z(x)=(1+x^2)eta((x-e)^2+eta^2)^{-1}$
-extends continuously to $overlineR$ with $k_z(∞)=eta$, and its
-integral against $sigma_r$ is $ImTr(H_r-z)^{-1}$; so any subsequential
-limit $sigma$ has the law of $Im m$ at a countable dense set of $z$,
-whence at all $z$ by continuity. For $fin C_c(R)$, integrating $k_z$
-against $pi^{-1}f(e)dd e$ and letting $eta↓0$ gives $(1+x^2)f(x)$
-on $R$ (Poisson identity), with uniformly bounded integrals and value
-$etapi^{-1}int fto0$ at infinity, so dominated convergence applies;
-applying the same inversion to $Im m(z)$ gives $int f,ddnu$. Every
-subsequential local vague limit of $nu^{(r)}$ therefore has the law of
-$nu$.
+For the point measures, let nu^{(r)} be the eigenvalue counting measure
+of H_r and sigma_r(dd x)=nu^{(r)}(dd x)/(1+x^2); its total mass
+ImTr(H_r-mathrm i)^rmi)^{-1} is tight by (eq:sw-trace-limit), so
+sigma_r is tight on the one-point compactification overlineR. For
+z=e+mathrm irmieta the kernel k_z(x)=(1+x^2)eta((x-e)^2+eta^2)^{-1}
+extends continuously to overlineR with k_z(∞)=eta, and its
+integral against sigma_r is ImTr(H_r-z)^{-1}; so any subsequential
+limit sigma has the law of Im m at a countable dense set of z,
+whence at all z by continuity. For fin C_c(R), integrating k_z
+against pi^{-1}f(e)dd e and letting eta↓0 gives (1+x^2)f(x)
+on R (Poisson identity), with uniformly bounded integrals and value
+etapi^{-1}int fto0 at infinity, so dominated convergence applies;
+applying the same inversion to Im m(z) gives int f,ddnu. Every
+subsequential local vague limit of nu^{(r)} therefore has the law of
+nu.
@@ -2375,141 +2365,141 @@ noise. We now apply fixed-energy universality to these finite matrices.
[In plain words]
The switching step reduces the microscopic limit to a finite
-matrix $H_r=$ diagonal $+$ GOE noise. To identify its eigenvalue law, we
+matrix H_r= diagonal + GOE noise. To identify its eigenvalue law, we
apply fixed-energy Dyson Brownian motion universality
(Landon--Sosoe--Yau): a diagonal matrix perturbed by a small GOE noise
has, in the bulk, the same local eigenvalue statistics as pure GOE,
provided the free-convolution comparison density at the origin is
-$1/pi$. Lemma~dbm:regularity-density verifies the regularity
-hypotheses, and the $c_r$ dilation adjusts that density to exactly
-$1/pi$.
+1/pi. Lemma~dbm:regularity-density verifies the regularity
+hypotheses, and the c_r dilation adjusts that density to exactly
+1/pi.
-For a real diagonal $V=diag(v_1,...,v_m)$ write
-$m_V(zeta)=m^{-1}sum_j(v_j-zeta)^{-1}$. For $g,G>0$ call $V$
-$(g,G)$-regular if, for fixed $c,C,C_V>0$,
+For a real diagonal V=diag(v_1,...,v_m) write
+m_V(zeta)=m^{-1}sum_j(v_j-zeta)^{-1}. For g,G>0 call V
+(g,G)-regular if, for fixed c,C,C_V>0,
cleIm m_V(e+mathrm irmieta)le Cquad (|e|le G, gleetale10),
- qquad |V|le m^{C_V},
+ |V|le m^{C_V},
-with $g,G$ possibly $m$-dependent and, for a fixed $delta>0$,
+with g,G possibly m-dependent and, for a fixed delta>0,
- m^{delta-1}le gle m^{-delta},qquad Gle m^{-delta}.
+ m^{delta-1}le gle m^{-delta}, Gle m^{-delta}.
-Let $W_m$ be standard normalized GOE (off-diagonal variance $1/m$,
-diagonal $2/m$). The free-convolution comparison density associated with
-$V+sqrt t,W_m$ is described by
+Let W_m be standard normalized GOE (off-diagonal variance 1/m,
+diagonal 2/m). The free-convolution comparison density associated with
+V+sqrt t,W_m is described by
- m_{{rm fc},t}(zeta)=m_V(zeta+t,m_{{rm fc},t}(zeta)),qquad
+ m_{{rm fc},t}(zeta)=m_V(zeta+t,m_{{rm fc},t}(zeta)),
rho_{{rm fc},t}(e)=pi^{-1}lim_{eta↓0}Im m_{{rm fc},t}(e+mathrm irmieta),
-using the upper-half-plane solution. Write $p_H^{(m)}$ for the unit-mass
-symmetrized joint eigenvalue density of $H$, and $p_H^{(a)}$ for its
-unit-mass $a$-point marginal.
+using the upper-half-plane solution. Write p_H^{(m)} for the unit-mass
+symmetrized joint eigenvalue density of H, and p_H^{(a)} for its
+unit-mass a-point marginal.
[Fixed-energy DBM input]
-Suppose a sequence of deterministic diagonal $V$ of dimension $m$
+Suppose a sequence of deterministic diagonal V of dimension m
satisfies (eq:dbm-regularity) and (eq:dbm-parameter-domain)
-with fixed constants. Fix $sigma>0$ and
+with fixed constants. Fix sigma>0 and
g,m^{sigma}le tle m^{-sigma}G^2,
-and assume $rho_{{rm fc},t}(0)=1/pi$. Then for every fixed $age1$ and
-$Oin C_c^∞(R^a)$ the difference of
-$int O(boldsymbolalpha),p^{(a)}_{V+sqrt tW_m}(piboldsymbolalpha/m)ddboldsymbolalpha$
-and $int O(boldsymbolalpha),p^{(a)}_{W_m}(piboldsymbolalpha/m)ddboldsymbolalpha$
-tends to zero. Consequently the expected sums of $O$ over ordered tuples
+and assume rho_{{rm fc},t}(0)=1/pi. Then for every fixed age1 and
+Oin C_c^∞(R^a) the difference of
+int O(boldsymbolalpha),p^{(a)}_{V+sqrt tW_m}(piboldsymbolalpha/m)ddboldsymbolalpha
+and int O(boldsymbolalpha),p^{(a)}_{W_m}(piboldsymbolalpha/m)ddboldsymbolalpha
+tends to zero. Consequently the expected sums of O over ordered tuples
of distinct points of the two eigenvalue processes, both rescaled by
-$m/pi$, differ by $o(1)$.
+m/pi, differ by o(1).
Proof.
This is Theorem~2.2 of Landon--Sosoe--Yau (with Definition~2.1 and
Equations~(2.5)--(2.9) there), specialized to energy zero and equal
densities. For the last assertion multiply each integral by
-$(m)_a/(m/pi)^atopi^a$, the Jacobian and ordered-tuple factor for the
+(m)_a/(m/pi)^atopi^a, the Jacobian and ordered-tuple factor for the
unit-mass convention.
-Fix a microscopic subsequential law $nu$ from
+Fix a microscopic subsequential law nu from
Proposition~prop:microscopic-limit and the parameters and
conditional matrices of Proposition~prop:gaussian-insertion:
-$K=r^{1/100}$, $T=r^{tau}$ with $0<tau<chi/100000$,
-$I={j:|x_j|le K}$, $q_*=|I|$, and
+K=r^{1/100}, T=r^{tau} with 0<tau<chi/100000,
+I={j:|x_j|le K}, q_*=|I|, and
- H_r=D_I+s_1Id+√(s_2),mathcal W_r,calW_r,qquad |s_1|le CT, cTle s_2le CT,
+ H_r=D_I+s_1Id+√(s_2),mathcal W_r,calW_r, |s_1|le CT, cTle s_2le CT,
-with $mathcal W_r$calW_r$ raw GOE of dimension $q_*$. By (eq:discrepancy),
+with mathcal W_rcalW_r raw GOE of dimension q_*. By (eq:discrepancy),
- q_*=2rho K+O(K^{1-chi}),qquad T=q_*^{100tau+o(1)}.
+ q_*=2rho K+O(K^{1-chi}), T=q_*^{100tau+o(1)}.
-Let $alpha_*=pirho$. Scaling by $alpha_*/q_*$,
+Let alpha_*=pirho. Scaling by alpha_*/q_*,
(alpha_*)/(q_*)H_rstackrel{law}{=}V_*+√(t_*),W_{q_*},
- qquad V_*=(alpha_*)/(q_*)(D_I+s_1Id),quad t_*=(alpha_*^2s_2)/(q_*).
+ V_*=(alpha_*)/(q_*)(D_I+s_1Id), t_*=(alpha_*^2s_2)/(q_*).
-For almost every $nu$, $V_*$ is eventually regular with
-$g=T^{1/2}/q_*$, $G=q_*^{-1/8}$, satisfying
+For almost every nu, V_* is eventually regular with
+g=T^{1/2}/q_*, G=q_*^{-1/8}, satisfying
(eq:dbm-parameter-domain) and (eq:dbm-time-window) with
-$m=q_*$, $delta=sigma=tau$. Moreover
-$rho_{{rm fc},t_*}(0)to1/pi$; with $c_r=pirho_{{rm fc},t_*}(0)$ the
-data $c_rV_*$, $c_r^2t_*$ satisfy Theorem~thm:dbm-input, including
+m=q_*, delta=sigma=tau. Moreover
+rho_{{rm fc},t_*}(0)to1/pi; with c_r=pirho_{{rm fc},t_*}(0) the
+data c_rV_*, c_r^2t_* satisfy Theorem~thm:dbm-input, including
the exact density condition.
Proof.
-$|V_*|$ is bounded since $T=o(K)$. Apply a slightly enlarged domain
-$|Rezeta|le2G$, $g/2leImzetale20$; the normalized resolvent trace is
-$m_{V_*}(zeta)=alpha_*^{-1}sum_{jin I}(x_j-(q_*zeta/alpha_*-s_1))^{-1}$.
-Writing $e=q_*Rezeta/alpha_*-s_1$, $u=q_*Imzeta/alpha_*$ gives
+|V_*| is bounded since T=o(K). Apply a slightly enlarged domain
+|Rezeta|le2G, g/2leImzetale20; the normalized resolvent trace is
+m_{V_*}(zeta)=alpha_*^{-1}sum_{jin I}(x_j-(q_*zeta/alpha_*-s_1))^{-1}.
+Writing e=q_*Rezeta/alpha_*-s_1, u=q_*Imzeta/alpha_* gives
- |e|=O(K^{7/8}+T)=o(K),qquad cT^{1/2}le ule CK.
+ |e|=O(K^{7/8}+T)=o(K), cT^{1/2}le ule CK.
-Equation~(eq:mesoscopic-limit) (radius comparable to $K$, fixed
-exponent below $50tau$) and the imaginary part of
+Equation~(eq:mesoscopic-limit) (radius comparable to K, fixed
+exponent below 50tau) and the imaginary part of
(eq:marked-representation) give
-$sum_j u((x_j-e)^2+u^2)^{-1}=pirho+o(1)$ uniformly in
-(eq:dbm-original-domain); terms with $|x_j|>K$ sum to $O(u/K)$, and
-the retained sum has positive lower bound $ge q_*u/(4K^2+u^2)ge c>0$ in
+sum_j u((x_j-e)^2+u^2)^{-1}=pirho+o(1) uniformly in
+(eq:dbm-original-domain); terms with |x_j|>K sum to O(u/K), and
+the retained sum has positive lower bound ge q_*u/(4K^2+u^2)ge c>0 in
the complementary regime. This gives regularity. The powers are
-$g=m^{-1+50tau+o(1)}$, $t_*=m^{-1+100tau+o(1)}$, $G=m^{-1/8}$, so
-$t_*/(gm^tau)=m^{49tau+o(1)}to∞$ and
-$t_*/(m^{-tau}G^2)=m^{-3/4+101tau+o(1)}to0$, verifying the time window
-and domain for small $tau$.
+g=m^{-1+50tau+o(1)}, t_*=m^{-1+100tau+o(1)}, G=m^{-1/8}, so
+t_*/(gm^tau)=m^{49tau+o(1)}to∞ and
+t_*/(m^{-tau}G^2)=m^{-3/4+101tau+o(1)}to0, verifying the time window
+and domain for small tau.
-For the density, $zeta=0$ in (eq:dbm-free-convolution) reads
+For the density, zeta=0 in (eq:dbm-free-convolution) reads
- w=F_r(w),qquad F_r(w)=alpha_*^{-1}sum_{jin I}(x_j-(alpha_*s_2w-s_1))^{-1}.
+ w=F_r(w), F_r(w)=alpha_*^{-1}sum_{jin I}(x_j-(alpha_*s_2w-s_1))^{-1}.
-On $|w-mathrm i|rmi|le1/4$ the argument $z=alpha_*s_2w-s_1$ has
-$Im zasymp T$, $|z|=O(T)$; (eq:mesoscopic-limit) and
+On |w-mathrm i|rmi|le1/4 the argument z=alpha_*s_2w-s_1 has
+Im zasymp T, |z|=O(T); (eq:mesoscopic-limit) and
(eq:marked-representation) give
-$PVsum_j(x_j-z)^{-1}=mathrm irmipirho+o(1)$, and
-$PVsum_{|x_j|>K}(x_j-z)^{-1}=O(K^{-chi}+T/K)$. Hence $F_r(w)=mathrm i+o(1)$rmi+o(1)$
-uniformly; Rouch'e for $w-F_r(w)$ against $w-mathrm i$rmi$ gives a unique
-simple zero $w_rtomathrm i$rmi$ in the disk, continued by the implicit
-function theorem to a positive-imaginary solution near $zeta=0$. That this
+PVsum_j(x_j-z)^{-1}=mathrm irmipirho+o(1), and
+PVsum_{|x_j|>K}(x_j-z)^{-1}=O(K^{-chi}+T/K). Hence F_r(w)=mathrm i+o(1)rmi+o(1)
+uniformly; Rouch'e for w-F_r(w) against w-mathrm irmi gives a unique
+simple zero w_rtomathrm irmi in the disk, continued by the implicit
+function theorem to a positive-imaginary solution near zeta=0. That this
is the physical branch follows from the identity
frac1{q_*}sum_{jin I}|(V_*)_{jj}-zeta-t_*w|^{-2}
=(Im w)/(Imzeta+t_*Im w)<frac1{t_*}
for every upper-half-plane solution; two distinct solutions would give,
-after subtracting and dividing, $1$ equal to a quantity of absolute value
+after subtracting and dividing, 1 equal to a quantity of absolute value
strictly less than one by Cauchy--Schwarz and
(eq:dbm-uniqueness-identity). Its boundary value gives
-$rho_{{rm fc},t_*}(0)=Im w_r/pito1/pi$.
+rho_{{rm fc},t_*}(0)=Im w_r/pito1/pi.
-Finally, replacing $V_*,t_*$ by $c_rV_*,c_r^2t_*$ dilates the
-free-convolution law by $c_r$, with transform
-$c_r^{-1}m_{{rm fc},t_*}(zeta/c_r)$ and density
-$c_r^{-1}rho_{{rm fc},t_*}(0)=1/pi$ at zero; since $c_rto1$ the enlarged
+Finally, replacing V_*,t_* by c_rV_*,c_r^2t_* dilates the
+free-convolution law by c_r, with transform
+c_r^{-1}m_{{rm fc},t_*}(zeta/c_r) and density
+c_r^{-1}rho_{{rm fc},t_*}(0)=1/pi at zero; since c_rto1 the enlarged
domain gives (eq:dbm-regularity) and the strict power margins
-preserve (eq:dbm-time-window) for $c_r^2t_*$.
+preserve (eq:dbm-time-window) for c_r^2t_*.
By (eq:dbm-normalized-matrix) the rescaled eigenvalues supplied by
@@ -2517,111 +2507,111 @@ Theorem~thm:dbm-input are
(q_*)/(pi)cdot(c_ralpha_*)/(q_*)lambda_j(H_r)=c_rrho,lambda_j(H_r),
-with $c_r=1$ at exceptional initial values.
+with c_r=1 at exceptional initial values.
-Let $mathcal G_m=calG_m=sum_{j=1}^mdelta_{(m/pi)lambda_j(W_m)}$. These
+Let mathcal G_m=calG_m=sum_{j=1}^mdelta_{(m/pi)lambda_j(W_m)}. These
converge vaguely in law along all dimensions to a common locally finite
-process $mathcal S_calS_{GOE}$: the eigenvalues of $sqrt m,W_m$ have
-density $propto e^{-frac14sum y_j^2}prod_{i<j}|y_i-y_j|$, and
-Theorem~1 of Valk'o--Vir'ag (for $beta=1$, center zero) gives vague
-convergence after multiplying by $2sqrt m$; dividing by $2pi$ gives
-$mathcal G_m$.calG_m$.
+process mathcal S_calS_{GOE}: the eigenvalues of sqrt m,W_m have
+density propto e^{-frac14sum y_j^2}prod_{i<j}|y_i-y_j|, and
+Theorem~1 of Valk'o--Vir'ag (for beta=1, center zero) gives vague
+convergence after multiplying by 2sqrt m; dividing by 2pi gives
+mathcal G_m.calG_m.
-For every compact $JsubsetR$ and every $b>0$,
-$sup_{mge1}EEexp(b,mathcal G_m(J))<∞$.calG_m(J))<∞$.
+For every compact JsubsetR and every b>0,
+sup_{mge1}EEexp(b,mathcal G_m(J))<∞.calG_m(J))<∞.
Proof.
Bound the GOE count by twice a Gaussian-unitary count plus one via the
-Gaussian decimation identity of Forrester--Rains. Let $mathrm O_m$rmO_m$ be the
+Gaussian decimation identity of Forrester--Rains. Let mathrm O_mrmO_m be the
decreasingly ordered points with density
-$proptoDelta(y)prod_i b_0(y_i)$, $b_0=e^{-y^2/2}$, the law of
-$sqrt m,W_m/sqrt2$; then $mathcal G_m(J)$calG_m(J)$ is the $mathrm O_m$rmO_m$ count in
-$J_m=pi J/√(2m)$, so $|J_m|=O(m^{-1/2})$. Superposing independent
-$mathrm O_m$rmO_m$ and $mathrm O_rmO_{m+1}$ gives density
-$P(z)=sum_{|S|=m}Delta(z_S)Delta(z_{S^c})$, which by the
+proptoDelta(y)prod_i b_0(y_i), b_0=e^{-y^2/2}, the law of
+sqrt m,W_m/sqrt2; then mathcal G_m(J)calG_m(J) is the mathrm O_mrmO_m count in
+J_m=pi J/√(2m), so |J_m|=O(m^{-1/2}). Superposing independent
+mathrm O_mrmO_m and mathrm O_rmO_{m+1} gives density
+P(z)=sum_{|S|=m}Delta(z_S)Delta(z_{S^c}), which by the
parity--Vandermonde factorization equals
P(z)=c_m,Delta(z_1,z_3,...,z_{2m+1}),Delta(z_2,z_4,...,z_{2m}),
-with $c_m>0$: every factor on the right divides $P$ (same-parity terms
+with c_m>0: every factor on the right divides P (same-parity terms
vanish at coincidences; the others cancel in pairs with sign ratio
-$(-1)^{j-i-1}=-1$), both sides have degree $m^2$, and the quotient is
+(-1)^{j-i-1}=-1), both sides have degree m^2, and the quotient is
positive. Integrating the odd points over their interlacing intervals with
-the monic polynomials $p_j=(-1)^jb_0^{(j)}/b_0$ (each row replaced by its
-partial row sums, last row $int b_0$, expansion along it) gives that the
+the monic polynomials p_j=(-1)^jb_0^{(j)}/b_0 (each row replaced by its
+partial row sums, last row int b_0, expansion along it) gives that the
even points have density
proptoDelta(y)^2prod_i e^{-y_i^2}.
-This density equals $(m!)^{-1}det(K_m(y_i,y_j))$ for the
-projection kernel $K_m=sum_{j=0}^{m-1}psi_jpsi_j$ of the orthonormal
-polynomials for $e^{-x^2}$, with order-$a$ factorial moment density
-$det(K_m(y_i,y_j))_{i,j=1}^a$. The operator $-∂_x^2+x^2$
-has eigenvalue $2j+1$ on $psi_j$, so $|f'|_2^2le(2m-1)|f|_2^2$ on the
-range of $K_m$, and $|f(x)|^2le2|f'|_2|f|_2$ gives
-$K_m(x,x)le2√(2m)$. Then for every $age0$,
-$EE(N_m^{even})_ale(2√(2m),|J_m|)^ale C_J^a$, so
-$EE e^{vN_m^{even}}leexp(C_J(e^v-1))$. As the
+This density equals (m!)^{-1}det(K_m(y_i,y_j)) for the
+projection kernel K_m=sum_{j=0}^{m-1}psi_jpsi_j of the orthonormal
+polynomials for e^{-x^2}, with order-a factorial moment density
+det(K_m(y_i,y_j))_{i,j=1}^a. The operator -∂_x^2+x^2
+has eigenvalue 2j+1 on psi_j, so |f'|_2^2le(2m-1)|f|_2^2 on the
+range of K_m, and |f(x)|^2le2|f'|_2|f|_2 gives
+K_m(x,x)le2√(2m). Then for every age0,
+EE(N_m^{even})_ale(2√(2m),|J_m|)^ale C_J^a, so
+EE e^{vN_m^{even}}leexp(C_J(e^v-1)). As the
superposition count is at most twice its even count plus one and dominates
-the $mathrm O_m$rmO_m$ count,
-$EE e^{bmathcal G_m(J)}calG_m(J)le e^bEE e^{2bN_m^{even}}$.
+the mathrm O_mrmO_m count,
+EE e^{bmathcal G_m(J)}calG_m(J)le e^bEE e^{2bN_m^{even}}.
-Let $nu^{(r)}=sum_jdelta_{lambda_j(H_r)}$; by
+Let nu^{(r)}=sum_jdelta_{lambda_j(H_r)}; by
Proposition~prop:gaussian-insertion,
nu^{(r)}⇒nu
-unconditionally in local vague topology. Fix $0le hin C_c^∞(R)$,
-$f=1-e^{-h}$, and for a finite point measure $zeta=sum_idelta_{y_i}$ set
-$A_0=1$, $A_a=(a!)^{-1}sum_{i_1,...,i_a distinct}prod_{j=1}^a f(y_{i_j})$,
-$B_ell=sum_{a=0}^{ell}(-1)^aA_a$. The Bonferroni inequalities give
+unconditionally in local vague topology. Fix 0le hin C_c^∞(R),
+f=1-e^{-h}, and for a finite point measure zeta=sum_idelta_{y_i} set
+A_0=1, A_a=(a!)^{-1}sum_{i_1,...,i_a distinct}prod_{j=1}^a f(y_{i_j}),
+B_ell=sum_{a=0}^{ell}(-1)^aA_a. The Bonferroni inequalities give
B_{2ell+1}(zeta)leexp(-int h,ddzeta)le B_{2ell}(zeta),
-seen by Bernoulli variables of success probabilities $f(y_i)$. Conditional
-on almost every $nu$, let $zeta_r$ have coordinates $c_rrho,lambda_j(H_r)$;
+seen by Bernoulli variables of success probabilities f(y_i). Conditional
+on almost every nu, let zeta_r have coordinates c_rrho,lambda_j(H_r);
Lemma~dbm:regularity-density and Theorem~thm:dbm-input with
-test $prod_j f(alpha_j)$ give, for each fixed $ell$,
-$EE[B_ell(zeta_r)midnu]-EE B_ell(mathcal G_calG_{q_*})to0$. To close
-the brackets uniformly in $q_*$, take $Jsupsetsupp h$; since
-$A_a(mathcal G_m)calG_m)lebinom{mathcal G_m(J)}calG_m(J)ale2^{-a}3^{mathcal G_m(J)}calG_m(J)$
-(from $sum_a2^abinom Na=3^N$), Lemma~lem:goe-counts gives
+test prod_j f(alpha_j) give, for each fixed ell,
+EE[B_ell(zeta_r)midnu]-EE B_ell(mathcal G_calG_{q_*})to0. To close
+the brackets uniformly in q_*, take Jsupsetsupp h; since
+A_a(mathcal G_m)calG_m)lebinom{mathcal G_m(J)}calG_m(J)ale2^{-a}3^{mathcal G_m(J)}calG_m(J)
+(from sum_a2^abinom Na=3^N), Lemma~lem:goe-counts gives
sup_mEE[B_{2ell}(mathcal G_m)-B_calG_m)-B_{2ell+1}(mathcal G_m)]calG_m)]le C_J2^{-(2ell+1)}.
-Using (eq:dbm-bonferroni) for both models, sending $rto∞$ at
-fixed $ell$ and then $ellto∞$, gives
+Using (eq:dbm-bonferroni) for both models, sending rto∞ at
+fixed ell and then ellto∞, gives
EEBigl[exp(-int h,ddzeta_r)midnuBigr]to
EEexp(-int h,ddmathcal S_calS_{GOE}).
-Both sides lie in $[0,1]$, so bounded convergence removes the conditioning.
-For $c>0$ let $mathcal D_ccalD_czeta$ scale positions by $c$; this map is
-continuous in $(c,zeta)$ for $c$ near a positive number and $zeta$
-locally vaguely convergent. Since $c_rto1$ almost surely,
-$zeta_r=mathcal D_calD_{c_rrho}nu^{(r)}⇒mathcal D_calD_rhonu$, and
+Both sides lie in [0,1], so bounded convergence removes the conditioning.
+For c>0 let mathcal D_ccalD_czeta scale positions by c; this map is
+continuous in (c,zeta) for c near a positive number and zeta
+locally vaguely convergent. Since c_rto1 almost surely,
+zeta_r=mathcal D_calD_{c_rrho}nu^{(r)}⇒mathcal D_calD_rhonu, and
bounded continuity of the Laplace functional of a compactly supported
continuous nonnegative test gives
EEexp(-int h,ddmathcal D_calD_rhonu)
=EEexp(-int h,ddmathcal S_calS_{GOE})
-for $0le hin C_c^∞$, extended to $0le hin C_c$ by convolution with
+for 0le hin C_c^∞, extended to 0le hin C_c by convolution with
a nonnegative approximate identity and bounded convergence.
Proof. [Proof of Theorem~thm:main]
Start with any sequence of admissible sizes; Proposition~prop:microscopic-limit
-supplies a further subsequence on which $nu_n⇒nu$ as above. The
-graph point measure is $mathcal D_calD_rhonu_n$, and by bounded continuity of
+supplies a further subsequence on which nu_n⇒nu as above. The
+graph point measure is mathcal D_calD_rhonu_n, and by bounded continuity of
its Laplace functional and (eq:dbm-limit-identification) its Laplace
-expectation along this subsequence tends to that of $mathcal S_calS_{GOE}$;
-the reference GOE expectations at dimension $n$ tend to the same value along
+expectation along this subsequence tends to that of mathcal S_calS_{GOE};
+the reference GOE expectations at dimension n tend to the same value along
all sizes. Every original subsequence thus has a further subsequence on
which the difference tends to zero, proving the full-sequence limit. All
excluded events had probability tending to zero in the original