goe_rrg_bulk_universality_v2.txt diff bc4e587..2d5618e

diff bc4e587..2d5618e goe_rrg_bulk_universality_v2.txt
diff --git a/goe_rrg_bulk_universality_v2.txt b/goe_rrg_bulk_universality_v2.txt
index 64c24c5..ebc58c6 100644
--- a/goe_rrg_bulk_universality_v2.txt
+++ b/goe_rrg_bulk_universality_v2.txt
@@ -32,32 +32,10 @@
 \DeclareMathOperator{\Ent}{Ent}
 \DeclareMathOperator{\Law}{Law}
 \setlength{\emergencystretch}{2em}
-\title{Bulk universality for fixed-degree random regular graphs}
-\author{The GOE prediction, proved; written to be read in full}
-\date{}
-
 \begin{document}
-\maketitle
-
-\begin{abstract}
-We prove the fixed-degree bulk universality conjecture for random regular
-graphs. Fix an integer $d\ge3$ and an energy $E$ strictly inside the
-limiting spectrum $(-2\sqrt{d-1},2\sqrt{d-1})$. Let $A_{n,d}$ be the
-adjacency matrix of the uniform simple labelled $d$-regular graph on $n$
-vertices. Zoom in on $E$ at the scale of one eigenvalue spacing, and
-unfold with the Kesten--McKay density $\rho_d$. Then the unfolded
-eigenvalue point process converges in law, along all admissible sizes,
-to the bulk point process of the Gaussian orthogonal ensemble. No
-conditioning on connectedness, bipartiteness, or any other graph
-property is imposed. The proof of Conjecture 2.11 of Bourgade--Huang is
-complete; every step that is not an imported theorem is proved here.
-\end{abstract}
 
-\tableofcontents
 
-\section{What is proved, and why it is surprising}\label{sec:intro}
 
-\subsection{The random object}
 
 Fix an integer $d\ge 3$. Let $G_{n,d}$ be the uniform random graph
 among all \emph{simple labelled} $d$-regular graphs on the vertex set
@@ -79,7 +57,6 @@ macroscopic scale. And yet, as we now explain, near any fixed interior
 energy the two matrices produce \emph{the same} local eigenvalue
 statistics. That is the surprise.
 
-\subsection{Macroscopic density}
 
 Kesten (1959) and McKay (1981) computed the limiting eigenvalue density
 of $G_{n,d}$: the empirical measure
@@ -106,7 +83,6 @@ only see averages over windows containing millions of eigenvalues. The
 interesting question is what happens at scale $1/n$, where individual
 eigenvalues are visible.
 
-\subsection{Local statistics: the statement}
 
 Fix \emph{once and for all} an energy
 \[
@@ -151,7 +127,6 @@ $1/\pi$. Following the same zoom-and-unfold recipe, define
   \Xi_n^{\mathrm{GOE}}:=\sum_{i=1}^n\delta_{(n/\pi)\mu_i}.
 \]
 
-\subsection{The theorem}
 
 \begin{theorem}[Main result]\label{thm:main}
 For every fixed integer $d\ge3$, every fixed energy $E$ in the open
@@ -199,7 +174,6 @@ $d\ge3$ is the honest hypothesis, and ``$d$ grows with $n$'' is a
 different (earlier) theorem.
 \end{itemize}
 
-\subsection{Why the prediction is striking}
 
 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
@@ -254,14 +228,13 @@ we use the switch symmetry together with the Gaussian marks to
 \emph{insert} a genuine Gaussian matrix perturbation, and the classical
 Dyson Brownian motion universality finishes the job.
 
-\subsection{How to read the proof: the four steps}
 
 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).
 
 \begin{enumerate}
-\item \textbf{Section~\ref{sec:entropy}: Gaussian moments at sampled
+\item \textbf{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.
@@ -270,21 +243,21 @@ 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 \textbf{Section~\ref{sec:limits}: endpoint resolvents and the
+\item \textbf{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 \textbf{Section~\ref{sec:switching}: continuous switching and
+\item \textbf{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 \textbf{Section~\ref{sec:dbm}: identification.} The diagonal
+\item \textbf{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
@@ -302,7 +275,6 @@ of microscopic width on the original graph; the microscopic information
 is produced by the entropy step and then transported by symmetry and
 Brownian motion.
 
-\subsection{Conventions}
 
 Throughout, $d\ge3$ and $E$ with $|E|<2\sqrt{d-1}$ are fixed, and every
 constant may depend on $d$ and $E$ without comment. We write $c,C$ for
@@ -315,7 +287,6 @@ eigenvectors are real eigenbasis vectors of $A_{n,d}$, normalized to
 have squared norm $n$; the choice inside an eigenspace is arbitrary
 and measurable.
 
-\section{Setting, tree identities, and the imported spectral input}\label{sec:prelim}
 
 This section fixes notation, records the three counting facts about the
 uniform regular-graph model that we use, states precisely the spectral
@@ -336,7 +307,6 @@ give the exact covariance of eigenvector values at nearby vertices; this
 is what makes the Gaussian endpoint marks appear later.
 \end{remark}
 
-\subsection{Microscopic coordinates and resolvents}
 
 For $\Im\zeta>0$ put
 \[
@@ -380,7 +350,6 @@ Theorem~\ref{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.
 
-\subsection{Channel matrices}
 
 For $k\ge1$ define
 \begin{equation}\label{eq:channel-matrices}
@@ -400,7 +369,6 @@ 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 $p$.
 
-\subsection{The uniform regular-graph model}
 
 We take graphs from the configuration (pairing) model and condition on
 simplicity. The facts we need are elementary and are proved here rather
@@ -474,7 +442,6 @@ vertex. A union bound over the fixed number of sampled edges gives the
 assertion.
 \end{proof}
 
-\subsection{The spectral estimates we import}
 
 Put
 \begin{equation}\label{eq:S}
@@ -550,7 +517,6 @@ one, and the observables in Theorem~\ref{thm:main} are bounded, so
 discarding vanishing events cannot change a limit.
 \end{remark}
 
-\subsection{Tree identities: covariance and logarithmic potential}
 
 The Gaussian-wave covariance on a regular tree is the object that
 records how eigenvector values at nearby vertices are correlated. We
@@ -687,7 +653,6 @@ by the substitution $t-E=\eta u$ and the integrability of
 $\log(1+u^{-2})$ at infinity.
 \end{proof}
 
-\section{Gaussian moments at sampled edges}\label{sec:entropy}
 
 The first of the four steps gives a fully quantitative answer to a
 qualitative question: do eigenvector values look Gaussian? The answer is
@@ -704,17 +669,16 @@ would not be enough is abstract weak convergence, which carries no rate.
 
 The strategy is the entropy method of Backhausz--Szegedy, made
 quantitative. A counting inequality for colourings of the graph
-(Section~\ref{sec:entropy-count}) bounds the number of labelings with
+(the coloured-star count) bounds the number of labelings with
 given local statistics. After a Gaussian smoothing, which turns the
 counting bound into a differential-entropy inequality, the deficit
 between the entropy of the star observation and half that of the edge
-observation is small. Section~\ref{sec:entropy-projection} removes the
+observation is small. The projection step removes the
 one nearly deterministic relation (the eigenvector equation), and
-Section~\ref{sec:entropy-tensor} shows that this small entropy deficit
+the tensor step shows that this small entropy deficit
 forces all Hermite moments of low degree to be small. That is Gaussian
 behaviour.
 
-\subsection{The statement}
 
 \begin{proposition}[Uniform endpoint moments]\label{prop:edge-moments}
 Fix positive integers $k,\ell$ and a nonnegative integer $q$. There are
@@ -756,7 +720,6 @@ every fixed polynomial moment matches, which is enough for everything
 we do later.
 \end{remark}
 
-\subsection{Observations, and the two entropies}\label{sec:entropy-count}
 
 Write
 \[
@@ -782,7 +745,6 @@ spelled out below.
 Positive polynomial error exponents may decrease from one occurrence to
 the next; this is harmless because only finitely many steps occur.
 
-\subsection{Counting coloured stars}\label{sec:entropy-counting}
 
 The first ingredient does not involve eigenvectors at all. Give the
 vertices of the configuration model labels from an alphabet of size $M$.
@@ -860,7 +822,6 @@ bounded factors. The statement does not refer to the stub order, so it
 passes to the uniform simple graph.
 \end{proof}
 
-\subsection{Smoothing, and the star--edge entropy inequality}\label{sec:entropy-smoothing}
 
 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
@@ -1035,7 +996,6 @@ regularization error $O(\eta)$ in turning $\log|t-E-\mathrm i\eta|$ into
 $\log|t-E|$. This proves \eqref{eq:ent-global-entropy}.
 \end{proof}
 
-\subsection{Removing the eigenvector constraint}\label{sec:entropy-projection}
 
 The entropy inequality \eqref{eq:ent-star-edge} still contains one
 nearly deterministic direction. An eigenvector satisfies
@@ -1044,7 +1004,7 @@ 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 Section~\ref{sec:entropy-tensor} needs.
+Hermite expansion in the tensor step needs.
 
 Write a star as $(Z,X)$ with $Z\in\R^{\ell}$ (the $\ell$ colour values
 at the centre) and $X\in\R^{d}\otimes\R^{\ell}$ (the colour values at
@@ -1231,7 +1191,6 @@ exactly the orthogonal map of $U_i$ that interchanges $w$ and $v_i$ and
 acts trivially on the colours. This proves the invariance assertion.
 \end{proof}
 
-\subsection{From an entropy deficit to Gaussian moments}\label{sec:entropy-tensor}
 
 We have the inequality \eqref{eq:ent-projected-entropy}, the approximate
 reversal symmetry, and the second moments. What is still missing is the
@@ -1422,11 +1381,10 @@ it is strictly positive on the unit sphere of that subspace. By
 compactness it is bounded below by $c_j\|H\|^2$ there.
 \end{proof}
 
-\subsection{The Gaussian criterion}\label{sec:entropy-criterion}
 
 We now extract Gaussian moments from the three ingredients above. The
 spaces $D$, $U_i$, projections $P_i$ and swaps $\sigma_i$ keep the
-geometry of Section~\ref{sec:entropy-projection}.
+geometry of the projection step.
 
 \begin{lemma}[Entropy criterion for Gaussian moments]\label{ent:gaussian-criterion}
 Fix an integer $q\ge2$. Let $B_n$ be random vectors in $D$ with
@@ -1607,7 +1565,7 @@ many $j\le q$ occur. This proves the lemma.
 \begin{proof}[Proof of Proposition~\ref{prop:edge-moments}]
 Condition on a graph in the common event of Lemma~\ref{ent:type-count}
 and Proposition~\ref{prop:spectral-input}, and let $B_n=B_V$, the
-projected star field of Section~\ref{sec:entropy-projection}. Complete
+projected star field of the projection step. Complete
 delocalization gives the deterministic bound $\|B_n\|\le n^{o(1)}$.
 Lemma~\ref{ent:projected-entropy} supplies the entropy hypothesis, the
 first two moments $\EE B_n=0$ and
@@ -1628,9 +1586,8 @@ the standard Gaussian $G$ along $w$ and $v_i$ form a $2\ell$-vector
 with covariance $C_\ell$, because $G$ has covariance $\Id_D$ in the
 $d\ell$ dimensions of $D$ and $\ip{w}{v_i}=p$. It remains to compare
 these coordinates with the actual edge values
-$\bigl(V(v),V(v')\bigr)\in\R^{\ell}\oplus\R^{\ell}$ of Section
-\ref{sec:entropy-count}. Write $B_V=(Z,\frac1{\sqrt a}PX)$ as in
-Section~\ref{sec:entropy-projection}. Then
+$\bigl(V(v),V(v')\bigr)\in\R^{\ell}\oplus\R^{\ell}$ of the coloured-star count. Write $B_V=(Z,\frac1{\sqrt a}PX)$ as in
+the projection step. Then
 \[
   \ip{v_i}{B_V}
   =p\ip{w}{B_V}+\ip{Pe_i}{PX}
@@ -1669,10 +1626,9 @@ and are the reason the proof strategy lets $n\to\infty$ before any of
 those parameters grow.
 \end{remark}
 
-\section{From smoothed counts to one energy}\label{sec:limits}
 
 We now have Gaussian moments at sampled edges. Two obstacles remain.
-First, all of Section~\ref{sec:entropy} concerns a graph on which we
+First, all of the entropy argument concerns a graph on which we
 already know the local law and rigidity; those are statements about
 \emph{smoothed} eigenvalue counts, and at the end we must pass to an
 unsmoothed statement at the single energy $E$. Second, moments of
@@ -1696,7 +1652,6 @@ 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.
 
-\subsection{The switch}
 
 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 \emph{switch} deletes the two
@@ -1790,7 +1745,6 @@ $u=e_v-e_y$ and $v=e_w-e_x$. Substituting the expressions for $X,Y$
 proves \eqref{eq:switch-entries}.
 \end{proof}
 
-\subsection{Why the switch is needed}\label{sec:limits-row}
 
 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
@@ -1802,7 +1756,6 @@ identity into a matrix relation among resolvent entries at the sampled
 endpoints. The next subsection extracts the scalar equation from that
 relation.
 
-\subsection{Resolvents at sampled edge endpoints}\label{sec:limits-endpoints}
 
 Conditionally on the graph, sample an infinite sequence of independent
 uniform oriented edges. For fixed $k\ge1$, let $W_{n,k}$ be the $n\times2k$
@@ -1929,7 +1882,6 @@ every step. Absorbing fixed constants into a smaller $\xi$ proves
 requirement $S\ge C(|s|+y+1)$, which holds before the limit.
 \end{proof}
 
-\subsection{The limiting marked process}\label{sec:limits-microscopic}
 
 For a locally finite point measure $\nu=\sum_j\delta_{x_j}$, write $\PV$
 for the symmetric truncation $\lim_{R\to\infty}\sum_{|x_j|\le R}$.
@@ -2082,7 +2034,6 @@ positive definite for every $z\in\C_+$, simultaneously over a countable
 set of $k$.
 \end{proof}
 
-\section{Continuous switches and Gaussian insertion}\label{sec:switching}
 
 We work with a microscopic subsequential law $\nu$ supplied by
 Proposition~\ref{prop:microscopic-limit}. Its endpoint marks are exactly
@@ -2091,7 +2042,7 @@ gives exact orthogonal switching symmetries; composing many small
 orthogonal switches then inserts a Gaussian matrix perturbation.
 
 \begin{remark}[In plain words]
-Section~\ref{sec:limits} leaves us with the microscopic limiting law and
+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
 the law. Composing many small conjugations then produces an honest GOE
@@ -2100,7 +2051,6 @@ 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.
 \end{remark}
 
-\subsection{From permutations to orthogonal switches}
 
 Fix $k\ge2$, write $M=M_k$, and for $O\in O(k)$ set
 $D_O=\begin{pmatrix}I_k&0\\0&O\end{pmatrix}$ and
@@ -2151,7 +2101,6 @@ orthogonal conjugates are all hyperplane reflections, and finite products
 generate $O(k)$.
 \end{proof}
 
-\subsection{The finite Gaussian model}
 
 Fix a discrepancy exponent $0<\chi<1/4$ from \eqref{eq:discrepancy}. Use
 $r$ planar rotations through a small angle $\theta$, with aggregate
@@ -2419,14 +2368,13 @@ subsequential local vague limit of $\nu^{(r)}$ therefore has the law of
 $\nu$.
 \end{proof}
 
-\section{Identification by finite-matrix Dyson Brownian motion}\label{sec:dbm}
 
 Proposition~\ref{prop:gaussian-insertion} approximates every microscopic
 subsequential law by the eigenvalues of a finite diagonal matrix plus GOE
 noise. We now apply fixed-energy universality to these finite matrices.
 
 \begin{remark}[In plain words]
-Section~\ref{sec:switching} reduces the microscopic limit to a finite
+The switching step reduces the microscopic limit to a finite
 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
@@ -2437,7 +2385,6 @@ hypotheses, and the $c_r$ dilation adjusts that density to exactly
 $1/\pi$.
 \end{remark}
 
-\subsection{The fixed-energy input}
 
 For a real diagonal $V=\diag(v_1,\dots,v_m)$ write
 $m_V(\zeta)=m^{-1}\sum_j(v_j-\zeta)^{-1}$. For $g,G>0$ call $V$
@@ -2485,7 +2432,6 @@ $(m)_a/(m/\pi)^a\to\pi^a$, the Jacobian and ordered-tuple factor for the
 unit-mass convention.
 \end{proof}
 
-\subsection{Regularity and the density adjustment}
 
 Fix a microscopic subsequential law $\nu$ from
 Proposition~\ref{prop:microscopic-limit} and the parameters and
@@ -2573,7 +2519,6 @@ Theorem~\ref{thm:dbm-input} are
 \end{equation}
 with $c_r=1$ at exceptional initial values.
 
-\subsection{The GOE limit and uniform count moments}
 
 Let $\mathcal G_m=\sum_{j=1}^m\delta_{(m/\pi)\lambda_j(W_m)}$. These
 converge vaguely in law along all dimensions to a common locally finite
@@ -2626,7 +2571,6 @@ the $\mathrm O_m$ count,
 $\EE e^{b\mathcal G_m(J)}\le e^b\EE e^{2bN_m^{\mathrm{even}}}$.
 \end{proof}
 
-\subsection{From correlation functions to Laplace functionals}
 
 Let $\nu^{(r)}=\sum_j\delta_{\lambda_j(H_r)}$; by
 Proposition~\ref{prop:gaussian-insertion},