diff eeeea42..b14a030 microstates_free_entropy_separation_v2.txt
diff --git a/microstates_free_entropy_separation_v2.txt b/microstates_free_entropy_separation_v2.txt
index 8f68b1f..b84208e 100644
--- a/microstates_free_entropy_separation_v2.txt
+++ b/microstates_free_entropy_separation_v2.txt
@@ -1,1020 +1,769 @@
-==============================================================================
- A FINITE-ENTROPY SEPARATION OF MICROSTATES AND
- NONMICROSTATES FREE ENTROPY
-==============================================================================
-
-PREFACE: HOW TO READ THIS PAPER
-
-Technical argument: high-school readers can follow the arc but not
-verify every line. In one sentence, chi and chi* are two information
-measures; chi counts matrix approximations, chi* integrates free
-Fisher information along semicircular noise; they usually agree, but
-we build X with both finite and chi(X) <= chi*(X) - 1/2.
-
-Proof idea in one paragraph. Small noise keeps chi(X) finite. Larger
-noise leaves a detectable block structure in the matrices, invisible
-to chi*. Data processing turns that structure into the gap.
-
-Guide. Section 1 is narrative; Sections 2-3 build tools; Sections 4-5
-build the example; Section 6 assembles the proof. Notation appears
-in-line; further symbols are introduced where first used.
-
-ABSTRACT
-
-We answer the finite-entropy equality question for microstates and
-nonmicrostates free entropy negatively. We construct a bounded
-self-adjoint tuple X in a von Neumann algebra with faithful normal
-tracial state such that
-
- -infinity < chi(X) <= chi*(X) - 1/2 < infinity.
-
-The construction uses a large but fixed number of variables.
-
-1. INTRODUCTION
-
-1.1 The two invariants.
-
-Fix a bounded self-adjoint tuple X = (X_1, ..., X_n) in a tracial von
-Neumann algebra (M, tau). Voiculescu attached to X two numerical
-invariants, both modelled on entropy.
-
-Microstates free entropy chi(X). For matrix size d, take the set of
-d x d self-adjoint n-tuples whose normalized trace of every short
-noncommutative word agrees with X to tolerance epsilon, and whose
-operator norms are bounded by a cutoff R. Its volume grows like
-exp(chi(X) d^2) up to polynomial factors. The precise definition is in
-Section 2.
-
-Nonmicrostates free entropy chi*(X). The free Fisher information
-Phi*(X) is the sum of squared L^2 norms of the conjugate variables of
-X, the free analogue of classical scores. Then chi* is an integral of
-Phi* along addition of standard semicircular noise. A standard
-semicircular variable is the free analogue of a centered Gaussian; the
-noise it generates plays the role of Brownian motion. The precise
-definition is in Section 2.
-
-For a single self-adjoint variable (n = 1), both invariants reduce to
-the same classical entropy.
-
-1.2 The question.
-
-Two facts were commonly expected:
- (a) chi(X) <= chi*(X) for every bounded self-adjoint X;
- (b) chi(X) > -infinity forces chi(X) = chi*(X).
-
-Fact (a) is a theorem. Biane-Capitaine-Guionnet proved it through
-large deviations for matrix Brownian motion. Jekel-Pi gave an
+Two invariants dominate the modern theory of free entropy. For a
+bounded self-adjoint tuple X = (X_1, ..., X_n) in a von Neumann algebra
+with a faithful normal tracial state tau, the microstates free entropy
+chi(X) is the asymptotic exponential volume, at matrix size d after
+dividing the logarithm by d^2, of self-adjoint d x d tuples whose
+normalized trace moments of short noncommutative words approximate those
+of X. The nonmicrostates free entropy chi*(X) integrates Voiculescu's
+free Fisher information Phi*(X) along free convolution with a standard
+semicircular family, the free analogue of integrating classical Fisher
+information along heat flow.
+
+Voiculescu introduced both quantities in a sequence of Inventiones
+papers hitting in 1991, 1994, and 1998, and consolidated the framework
+in a 2002 Bulletin of the London Mathematical Society survey that also
+posed the finiteness question that motivates the construction. Guionnet's Probability
+Surveys article collects the definitions, the standard comparison
+results, and the original form of the conjecture.
+
+Online (publisher): https://doi.org/10.1007/BF01245072
+Online (publisher): https://doi.org/10.1007/BF01231539
+Online (publisher): https://doi.org/10.1007/s002220050222
+Online (publisher): https://doi.org/10.1112/S0024609301008992
+Local PDF: ./p_guionnet-survey.pdf
+Online: https://www.emis.mi.sanu.ac.rs/EMIS/journals/PS/images/getdoc1af6.pdf?id=48&article=18&mode=pdf
+
+The inequality chi(X) <= chi*(X) holds for every bounded self-adjoint
+tuple. Biane, Capitaine, and Guionnet proved it via large deviation
+bounds for Hermitian matrix-valued Brownian motion; Jekel and Pi gave an
elementary proof via classical entropy and Fisher information, and
-extended it to conditional free entropy.
-
-Fact (b) was formulated by Guionnet (Conjecture 7.1 in her survey).
-The finiteness condition is a genuine hypothesis: Jekel-Pi recall
-examples with chi = -infinity < chi*. The open question was whether
-finiteness alone is enough.
-
-Our main result answers it in the negative.
-
-Theorem 1 (Main). There exist n >= 2 and a bounded self-adjoint tuple
-X = (X_1, ..., X_n) in a von Neumann algebra with a faithful normal
-tracial state such that
-
- -infinity < chi(X) <= chi*(X) - 1/2 < infinity.
-
-The number of variables n is large but fixed.
-
-1.3 Road map of the proof.
-
-Fix p large and put n = p + 1. Let B_1, ..., B_p be free variance-one
-semicirculars. In a separate tensor factor let Y be uniform on the p
-points y_l = -1 + 2(l-1)/(p-1), l = 1, ..., p. Then Y commutes with
-each B_i but is not free from them. Set
-
- C = (B_1, ..., B_p, Y),
- V_a = C + sqrt(a) S (S a standard semicircular n-tuple free from C),
- X = V_h (h > 0 small),
- L(a) = (1/2) sum_i log(2 pi e (a + c_i)) (c_i = tau(C_i^2)).
-
-Step A (finite chi; Lemma 4.2). chi(X) > -infinity: bounded matrix
-models of C, noised by Gaussian matrices of variance h, supply a
-positive-volume matching set.
-
-Step B (block event; Lemma 6.1). Evolved microstate ensembles of X
-retain a detectable simultaneous block structure in the last
-coordinate: probability at least 1/2; at most exp(-4 d^2) for the
-matching GUE reference.
-
-Step C (data processing). Both bounds give KL >= 2 d^2 in the limit.
-The GUE reference has explicit density, so
-
- limsup_d H_d(Z^{(d)}(t)) <= L(h+t) - 1.
-
-Step D (analytic chi*). For a > 0,
-
- chi*(V_a) = L(a) - (1/2) integral_a^infinity
- [ Phi*(V_u) - sum_i 1/(u + c_i) ] du.
-
-The Fisher deficit is bounded by squared higher-cumulant norms
-(Proposition 3.2); the signal's high cumulants are sparse (Lemma 4.1),
-so the deficit is small for large a.
-
-Step E (entropy production; Lemma 5.1). Jekel-Pi along the SAME
-ensemble:
-
- liminf_d H_d(Z^{(d)}(t))
- >= chi_R(X) + (1/2) integral_0^t Phi*(V_{h+s}) ds.
-
-Step F (assembly). Combine the two bounds on the same terminal law,
-using chi*(V_{h+t}) - chi*(V_h) = (1/2) int_0^t Phi*(V_{h+s}) ds:
-
- chi_R(X) <= L(h+t) - 1 - (1/2) int_0^t Phi*(V_{h+s}) ds
- <= chi*(V_h) - 1/2.
-
-Supremum over cutoffs R gives Theorem 1.
-
-1.4 Why two noise scales.
-
-The two bounds make competing scale demands.
- - Step D needs t >> sqrt(p), so the analytic deficit (order
- K^6 p^{3/2} t^{-3}) is small.
- - Step B needs t log(p) / p -> 0, so the unitary-net cost dominates
- Gaussian concentration in the block event.
-
-The single choice t = p^{3/4} satisfies both as p -> infinity. In
-explicit numbers, p = 2^64 and t = 2^48: then 2 K^2 sqrt(p) = 2^47 < t
-and 32 t / p = 2^{-11}. Choose p and t first; then take h small; then
-let matrix sizes d -> infinity.
-
-1.5 Relation to companion work.
-
-A companion paper proves each of the four free entropy dimensions of
-L(F_2) attains every integer value n >= 2 on some generating tuple.
-That statement varies the tuple inside a fixed algebra. Here we
-compare chi and chi* on a single fixed tuple, with both values finite.
-
-1.6 Organization.
-
- Section 2: conventions; Lemma 2.1 (variational formula for Phi*),
- Lemma 2.2 (bounded matrix inputs).
- Section 3: cumulants; Lemma 3.1 (Wick testing), Proposition 3.2
- (Fisher deficit bound).
- Section 4: the signal; Lemma 4.1 (cumulant bound), Lemma 4.2
- (chi(X) > -infinity).
- Section 5: entropy production; Lemma 5.1.
- Section 6: block event, Lemma 6.1, and proof of Theorem 1.
-
-
-==============================================================================
-2. CONVENTIONS AND PRELIMINARY FACTS
-==============================================================================
-
-We set up notation once and prove the two matrix-limit facts used in
-later sections. Section 2.1 fixes the algebra. Section 2.2 defines
-the microstates entropy chi. Section 2.3 defines conjugate variables
-and the free Fisher information Phi*. Section 2.4 proves a variational
-formula for Phi* (Lemma 2.1). Section 2.5 defines the nonmicrostates
-entropy chi*. Section 2.6 fixes matrix and GUE notation. Section 2.7
-proves the matrix convergence lemma (Lemma 2.2).
-
-2.1 Setting.
-
-Noncommutative variables live in a von Neumann algebra (M, tau) with a
-faithful normal tracial state tau. Write
-
- ||a||_2^2 = tau(a^* a).
-
-A standard semicircular family has variance one in each coordinate and
-is free. Every semicircular noise introduced below is free from the
-tuple it perturbs.
-
-2.2 Microstates entropy.
-
-Let X = (X_1, ..., X_n) be a bounded self-adjoint tuple. For R > 0,
-m >= 1, epsilon > 0, let Gamma_R(X; m, d, epsilon) be the set of
-self-adjoint d x d matrix tuples with coordinate operator norms at most
-R whose normalized trace of every word of length at most m differs from
-the value on X by less than epsilon. Take volume in the real Euclidean
-space of matrix tuples with inner product
-
- <A, B> = Re sum_{i=1}^n Tr(A_i^* B_i).
-
-Define
-
- chi_R(X) = inf_{m, epsilon} limsup_{d -> infinity}
- [ d^{-2} log Vol Gamma_R(X; m, d, epsilon)
- + (n/2) log d ],
- chi(X) = sup_{R > 0} chi_R(X).
-
-The additive (n/2) log d matches the classical entropy normalization
-below. If no matrices ever match X this way for a given R, the value
-is -infinity, which is allowed.
-
-2.3 Conjugate variables and free Fisher information.
-
-A bounded self-adjoint tuple Z has conjugate variables xi_1, ..., xi_n
-in L^2(M, tau) if for every noncommutative polynomial P in the
-indeterminates x_1, ..., x_n,
-
- tau(xi_i P(Z)) = (tau tensor tau)((partial_i P)(Z)),
-
-where the free difference quotient partial_i is linear and satisfies the
-Leibniz rule
-
- partial_i x_j = delta_ij (1 tensor 1),
- partial_i (P Q) = (partial_i P)(Q) + (P tensor 1)(partial_i Q),
-
-applied by substituting Z. Define the free Fisher information
-
- Phi*(Z) = sum_{i=1}^n ||xi_i||_2^2
+extended the inequality to conditional free entropy. Equality chi =
+chi* holds in several regular settings: Dabrowski's theorem covers a
+class of strictly convex potentials with Lipschitz derivatives and
+noncommutative polynomial approximation; Jekel's theorem covers convex
+matrix potentials with appropriate concentration and trace-polynomial
+approximation. These hypotheses rule out the geometric obstruction
+exploited below.
+
+Online (publisher): https://doi.org/10.1007/s00222-002-0281-4
+Local PDF: ./p_jekelpi.pdf
+Online: https://arxiv.org/pdf/2305.02574
+Local PDF: ./p_dabrowski.pdf
+Online: https://arxiv.org/pdf/1604.06420
+Local PDF: ./p_jekel.pdf
+Online: https://arxiv.org/pdf/1805.08814
+
+Finiteness of chi(X) does not force equality. Jekel and Pi recall
+nonapproximable laws with chi = -infinity < chi*. Guionnet formulated
+the remaining version of the question as Conjecture 7.1 of her survey:
+whether chi(X) > -infinity implies chi(X) = chi*(X). The result proved
+here answers it negatively:
+
+ -infinity < chi(X) <= chi*(X) - 1/2 < infinity
+
+for some bounded self-adjoint tuple X in a von Neumann algebra with a
+faithful normal tracial state. The tuple uses a large but fixed number
+of generators; no claim is made for small n or for tuples of independent
+interest.
+
+The construction runs on two scales of semicircular noise around a rough
+signal C built from p free semicirculars plus a commuting uniform
+variable Y. Tensor independence gives C sparse higher cumulants.
+Small noise X = C + sqrt(h) S keeps chi(X) finite. A larger noise
+V_a = C + sqrt(a) S, at a comparable to t = p^{3/4}, gives a controlled
+Fisher deficit at positive noise: the free Fisher information of V_a is
+close to that of a semicircular family of the same covariance, so chi*(V_a)
+sits just below the classical entropy L(a) of that family. Applying the
+Jekel-Pi comparison along the same evolved matrix ensemble and matching
+its terminal-time normalized entropy against the Gaussian reference of
+the same covariance yields the strict gap of 1/2. The ingredients are:
+
+ (i) two matrix convergence lemmas - a variational formula for Phi* and
+ uniform trace convergence for bounded deterministic input;
+
+ (ii) a cumulant-Fisher estimate reducing the deficit of Phi* against
+ the semicircular baseline to squared higher-cumulant norms;
+
+ (iii) the specific signal C and its sparse cumulants;
+
+ (iv) a de Bruijn identity for the evolved ensemble, whose Fisher
+ information dominates Phi*(V_{h+s}) in the matrix limit;
+
+ (v) a block event - the persistence of a simultaneous p-block
+ decomposition of the last coordinate at cutoff R - whose
+ probability stays above 1/2 under the evolved law and below
+ exp(-4 d^2) under the matching Gaussian reference;
+
+ (vi) an entropy sandwich that closes the argument.
+
+A companion result, established elsewhere, resolves the free group
+factor isomorphism problem affirmatively and shows each of the four free
+entropy dimensions of L(F_2) attains every integer value n >= 2 on
+suitable generating tuples. That statement varies the tuple inside a
+fixed von Neumann algebra; the construction fixes a tuple and
+compares chi and chi* on it, both finite.
+
+Local PDF: ./p_freegroup.pdf
+
+The quantitative definitions are as follows. Let m >= 1 be an integer,
+epsilon > 0, R > 0 a cutoff, and d a matrix dimension. Write
+Gamma_R(X; m, d, epsilon) for the set of self-adjoint d x d matrix
+tuples (A_1, ..., A_n) with coordinate operator norms at most R and with
+the normalized trace of every word of length at most m within epsilon of
+its value on X. Volume is taken in the real Euclidean space of tuples
+with inner product <A, B> = Re sum_i Tr(A_i^* B_i). Then
+
+ chi_R(X) = inf_{m, epsilon} limsup_{d -> infinity}
+ [ d^{-2} log Vol Gamma_R(X; m, d, epsilon) + (n/2) log d ],
+ chi(X) = sup_R chi_R(X),
+
+with the convention that chi is -infinity when every chi_R is. The
+additive (n/2) log d matches the classical entropy normalization below.
+
+Conjugate variables live in L^2(M, tau). For a noncommutative polynomial
+P in n formal indeterminates, the free difference quotient partial_i is
+the linear map satisfying partial_i x_j = delta_{ij} (1 tensor 1) and
+the Leibniz rule. Given a bounded self-adjoint tuple Z, the conjugate
+variables xi_1, ..., xi_n are the unique elements of L^2 satisfying
+
+ tau(xi_i P(Z)) = (tau tensor tau)((partial_i P)(Z))
+
+for every polynomial P. Put Phi*(Z) = sum_i ||xi_i||_2^2, possibly
+infinite when conjugate variables fail to exist. That conjugate
+variables are self-adjoint and unique in L^2 is standard.
+
+Local PDF: ./p_nica-speicher.pdf
+Online: https://rolandspeicher.files.wordpress.com/2020/06/nica-speicher-book.pdf
+
+Lemma 1 (Variational formula for Phi*). If Z has conjugate variables
+xi_i in L^2, then
+
+ Phi*(Z) = sup sum_i [ 2 (tau tensor tau)((partial_i P_i)(Z)) - tau(P_i(Z)^2) ],
+
+the supremum over self-adjoint noncommutative polynomials P_1, ..., P_n.
+The supremum may be restricted to polynomials with coefficients in Q + iQ.
+
+Proof. Polynomial density in L^2. The closed span K of {Q(Z) : Q
+polynomial} is invariant under each bounded self-adjoint Z_j, because
+Z_j Q(Z) is again a polynomial in Z. The orthogonal projection onto K
+commutes with Z_j, hence with W^*(Z). Since it fixes 1, it fixes every
+bounded algebra vector; those are dense in L^2. So K = L^2.
-when the xi_i exist, and Phi*(Z) = infinity otherwise. Conjugate
-variables are unique and self-adjoint when they exist.
+Self-adjointness. The identity partial_i(P^*) = flip((partial_i P)^*)
+together with symmetry of tau tensor tau shows xi_i^* is also a
+conjugate variable. Uniqueness from L^2 density makes xi_i self-adjoint.
-2.4 A variational formula for Phi*.
+Variational identity. For self-adjoint P_i,
-Lemma 2.1 (Polynomial variational formula). If a bounded self-adjoint
-Z has conjugate variables, then
+ sum_i [ 2 tau(xi_i P_i(Z)) - ||P_i(Z)||_2^2 ]
+ = sum_i [ ||xi_i||_2^2 - ||xi_i - P_i(Z)||_2^2 ],
- Phi*(Z) = sup sum_{i=1}^n ( 2 (tau tensor tau)((partial_i P_i)(Z))
- - tau(P_i(Z)^2) ), (2.1)
+so the supremum recovers sum_i ||xi_i||_2^2. Restricting the supremum
+to rational coefficients and self-adjoint P_i preserves the value,
+because Z has bounded moments. QED.
-the supremum being over self-adjoint noncommutative polynomials
-P_1, ..., P_n. It suffices to take P_i = (Q_i + Q_i^*)/2 with Q_i
-having coefficients in Q + i Q. Consequently, along any family of
-tuples with continuous moments and existing conjugate variables, Phi*
-is a measurable function of the parameter.
-
-Proof. Self-adjointness. The identity
-partial_i(P^*) = flip((partial_i P)^*) together with symmetry of
-tau tensor tau shows xi_i^* is also a conjugate variable. Uniqueness
-from polynomial L^2 density (below) makes xi_i = xi_i^*.
-
-Variational identity. The expression under the supremum equals
-
- sum_i [ 2 tau(xi_i P_i(Z)) - ||P_i(Z)||_2^2 ]
- = sum_i [ ||xi_i||_2^2 - ||xi_i - P_i(Z)||_2^2 ],
-
-so the supremum over all self-adjoint polynomials recovers
-sum_i ||xi_i||_2^2.
-
-Polynomial density. Let K be the closed span in L^2 of 1 and the
-polynomial vectors. Each bounded self-adjoint Z_j preserves K, so the
-orthogonal projection onto K commutes with Z_j and with W^*(Z). Since
-it fixes 1, it fixes every bounded algebra vector, and those are dense
-in L^2. Hence K = L^2.
-
-Countable and measurable versions. Symmetrizing P_i via
-(Q_i + Q_i^*)/2 gives the self-adjoint supremum. Rational
-coefficients give a countable dense subset. Each functional in the
-countable supremum is continuous in the moments of Z, so Phi* is
-measurable in the parameter. QED.
-
-2.5 Nonmicrostates entropy.
-
-Our nonmicrostates normalization is, for S a standard semicircular
+The nonmicrostates normalization is, for S a standard semicircular
n-tuple free from Z,
- chi*(Z) = (n/2) log(2 pi e)
- + (1/2) integral_0^infinity
- [ n / (1 + s) - Phi*(Z + sqrt(s) S) ] ds. (2.2)
+ chi*(Z) = (n/2) log(2 pi e)
+ + (1/2) integral_0^infinity
+ [ n / (1 + s) - Phi*(Z + sqrt(s) S) ] ds.
-The n / (1 + s) is the free Fisher integrand of n free semicircular
-coordinates; for a semicircular family the bracket vanishes identically
-and chi* equals the free entropy of that family, namely
-(n/2) log(2 pi e sigma^2).
+For a semicircular family of variances w_i the bracket vanishes and
+chi* reduces to (n/2) sum_i log(2 pi e w_i), matching the free entropy
+of that family. For general Z the integral is finite on every compact
+interval of positive noise times once Phi*(Z + sqrt(s) S) <= n/s, an
+inequality proved below (Lemma 3).
-2.6 Matrix notation and GUE.
+Matrix conventions. Write tr_d = d^{-1} Tr and
+||D||_{2,d}^2 = sum_i tr_d(D_i^* D_i) for a matrix tuple D. A standard
+GUE tuple G^{(d)} consists of n independent centered Gaussian
+self-adjoint matrices whose real Hilbert-Schmidt coordinates are
+independent with variance 1/d. For a random matrix tuple with density f,
+the normalized classical entropy is
-Write tr_d = d^{-1} Tr and, for a matrix tuple D,
-
- ||D||_{2,d}^2 = sum_{i=1}^n tr_d(D_i^* D_i).
-
-A standard GUE tuple G^{(d)} consists of independent centered Gaussian
-matrices whose real Hilbert-Schmidt coordinates are independent with
-variance 1/d. For a random matrix tuple with density f, its normalized
-classical entropy is
-
- H_d = -(1/d^2) integral f log f + (n/2) log d.
+ H_d = -(1/d^2) integral f log f + (n/2) log d.
Independent GUE coordinates of variances w_i > 0 have
+H_d = (1/2) sum_i log(2 pi e w_i), matching the microstate normalization.
- H_d = (1/2) sum_i log(2 pi e w_i).
-
-This matches the microstates normalization exactly.
+Lemma 2 (Bounded matrix inputs). Let K_d be sets of self-adjoint matrix
+tuples with a common coordinate operator-norm bound. Suppose the
+normalized trace moments converge uniformly over K_d to those of a
+bounded tuple T. Let A^{(d)} be a random tuple supported on K_d,
+independent of a fixed finite standard GUE tuple G^{(d)}. Then:
-2.7 Matrix convergence under bounded input.
+ (i) every normalized polynomial trace in (A^{(d)}, G^{(d)}) converges
+ in probability to its value at (T, S), where S is a standard
+ semicircular n-tuple free from T;
-The classical input to Lemma 2.2 is Voiculescu's asymptotic freeness:
-if deterministic matrix tuples converge in normalized trace moments,
-adjoining independent GUE matrices makes them converge to free products
-with semicirculars. Lemma 2.2 supplies the extra probability and
-expectation statements we need, including a uniform version over
-microstate sets.
-
-Lemma 2.2 (Bounded matrix inputs). Let K_d be sets of self-adjoint
-matrix tuples with a common coordinate operator-norm bound. Suppose
-their normalized trace moments converge uniformly over K_d to the
-moments of a bounded tuple T. Let A^{(d)} be any random tuple supported
-on K_d, independent of a fixed finite standard GUE tuple G^{(d)}. Then
-
- (i) every normalized polynomial trace in (A^{(d)}, G^{(d)})
- converges in probability to its value at (T, S), where S is a
- standard semicircular family free from T;
(ii) products of finitely many such traces converge in expectation;
- (iii) the dimensions may run over any sequence tending to infinity.
-Proof.
+ (iii) the dimensions may range over any sequence tending to infinity.
-GUE norm bound. A 1/4-net of the unit sphere of C^d in real dimension
-2d has cardinality at most 9^{2d}. For Hermitian G,
+Proof. GUE norm bound. A 1/4-net of the unit sphere of C^d, viewed in
+real dimension 2d, has cardinality at most 9^{2d}. For Hermitian G,
- ||G|| <= 2 max_{v in net} |v^* G v|.
+ ||G|| <= 2 max_{v in net} |v^* G v|.
-For standard GUE, each v^* G v is a real centered Gaussian of variance
-1/d because Tr((v v^*)^2) = 1. Therefore
+For standard GUE, each v^* G v is centered real Gaussian of variance
+1/d because Tr((v v^*)^2) = 1. So
- P( ||G|| > L ) <= 2 exp( d (2 log 9 - L^2 / 8) ). (2.3)
+ P(||G|| > L) <= 2 exp(d (2 log 9 - L^2/8)). (1)
-In particular, the coordinate norms of any fixed finite GUE tuple are
-at most 8 with probability tending to one; integrating the tail gives
-uniformly bounded moments of ||G||.
+In particular the coordinate norms of any fixed finite GUE tuple are at
+most 8 with probability tending to one, and every fixed moment of
+||G|| is uniformly bounded in d.
-Gradient bound. Let a be a deterministic tuple with coordinate norms
-at most R and put F = tr_d P(a, G) for a fixed polynomial P. Let
-(E_alpha) be a Hermitian basis orthonormal for the unnormalized
-Hilbert-Schmidt inner product and write G_j = sum_alpha g_{j,alpha}
-E_alpha. The g_{j,alpha} are independent Gaussians of variance 1/d.
+Gradient bound. Fix a deterministic tuple a of coordinate norms at
+most R and put F = tr_d P(a, G) for a fixed polynomial P. Choose a
+Hermitian basis E_alpha orthonormal for the unnormalized Hilbert-Schmidt
+inner product, and write G_j = sum_alpha g_{j alpha} E_alpha. The
+g_{j alpha} are independent centered Gaussians of variance 1/d.
Cyclicity of trace gives
- partial_{g_{j,alpha}} F = d^{-1} Tr(E_alpha B_j),
- sum_{j,alpha} |partial_{g_{j,alpha}} F|^2
- = d^{-1} sum_j tr_d(B_j^* B_j),
+ partial_{g_{j alpha}} F = d^{-1} Tr(E_alpha B_j),
+ sum_{j, alpha} |partial_{g_{j alpha}} F|^2 = d^{-1} sum_j tr_d(B_j^* B_j),
where B_j is the sum of cyclically ordered products obtained from P by
deleting each occurrence of the j-th GUE letter.
-Hermite expansion. A polynomial in standard real Gaussian coordinates
-has variance at most the expected squared gradient: each differentiation
-weights the squared modulus of a nonconstant Hermite coefficient by its
-positive total degree. Rescaling to variance 1/d and applying the same
-to real and imaginary parts:
+Hermite variance bound. A polynomial in standard real Gaussian
+coordinates has variance at most the expected squared gradient:
+differentiation weights each nonconstant Hermite coefficient's squared
+modulus by its positive total degree. After rescaling to variance 1/d
+and applying the same to real and imaginary parts,
- E |F - E F|^2 <= d^{-2} sum_j E tr_d(B_j^* B_j)
- = O_{P,R}(d^{-2}).
+ E |F - E F|^2 <= d^{-2} sum_j E tr_d(B_j^* B_j) = O_{P, R}(d^{-2}).
-So for every deterministic input sequence a_d converging in moments,
-F converges in probability to its value at (T, S). Voiculescu's
-trace-convergence theorem upgrades the expectation convergence to
-convergence in probability for the limit.
-
-Uniformity over K_d. Suppose uniformity failed: then for some
-polynomial and some threshold eta > 0, there existed a subsequence of
-dimensions and points a_d in K_d with error probability at least eta.
-Those a_d form a bounded deterministic moment-convergent sequence,
-contradicting the previous paragraph. Conditioning on A^{(d)} gives
-convergence in probability for the random input A^{(d)}.
+Convergence. So F converges in probability on every deterministic
+moment-convergent sequence a_d, with expectation converging to the value
+at (T, S) by Voiculescu's trace-convergence theorem. Uniformity over
+K_d follows: a failure would produce a subsequence d_k and points
+a_{d_k} in K_{d_k} whose error probabilities are bounded away from
+zero, forming a bounded deterministic moment-convergent sequence that
+contradicts the previous sentence. Conditioning on A^{(d)} gives
+convergence in probability for the random input.
Expectations. For a fixed product Q of normalized polynomial traces,
- |Q(A^{(d)}, G^{(d)})| <= C (1 + max_j ||G_j^{(d)}||)^D
-
-with C, D depending only on the polynomials and the common cutoff.
-Thus E |Q|^2 is uniformly bounded, Q is uniformly integrable, and
-convergence in probability implies convergence of expectations. QED.
+ |Q(A^{(d)}, G^{(d)})| <= C (1 + max_j ||G_j^{(d)}||)^D,
+with C, D depending only on the polynomials and the common cutoff. The
+squared estimate is uniformly integrable by the norm-moment bound, so
+convergence in probability gives convergence of expectations. QED.
-==============================================================================
-3. A CUMULANT BOUND FOR THE FISHER DEFICIT
-==============================================================================
+Free cumulants are the standard bookkeeping device. For a centered
+bounded self-adjoint n-tuple C, let kappa_m(C) denote the order-m free
+cumulant array, with
-Goal of this section: bound how far Phi*(V_u) is from sum_i 1 / (u + c_i),
-the value for a semicircular family of the same covariance. The bound
-is in terms of squared higher-cumulant norms of the signal C. Section
-3.1 recalls free cumulants. Section 3.2 tests higher-cumulant terms
-against semicircular Wick polynomials. Section 3.3 assembles the bound
-(Proposition 3.2).
+ ||kappa_m(C)||_2^2 = sum_{i_1, ..., i_m} |kappa_m(C_{i_1}, ..., C_{i_m})|^2.
-3.1 Free cumulants.
+Three facts enter: the moment-cumulant formula and its Mobius inversion
+on noncrossing partitions; vanishing of mixed cumulants between free
+algebras; and that a standard semicircular family has only second
+cumulants, given by its covariance. All three are in Nica-Speicher.
-For a centered bounded self-adjoint n-tuple C, let kappa_m(C) be its
-order-m free cumulant array with
+Local PDF: ./p_nica-speicher.pdf
+Online: https://rolandspeicher.files.wordpress.com/2020/06/nica-speicher-book.pdf
- ||kappa_m(C)||_2^2
- = sum_{i_1, ..., i_m = 1}^n |kappa_m(C_{i_1}, ..., C_{i_m})|^2.
+Lemma 3 (Wick testing). Let C be a centered bounded self-adjoint
+n-tuple, S a standard semicircular n-tuple free from C, and
+V_u = C + sqrt(u) S for u > 0. For a multi-index j = (j_1, ..., j_r)
+with r >= 1 and a monomial P in x_1, ..., x_n, set
-We use three standard facts: the moment-cumulant formula and its Mobius
-inversion on noncrossing partitions; vanishing of mixed cumulants
-between free algebras; semicirculars have only second cumulants.
+ D_j(P) = sum_{P = P_0 x_{j_1} P_1 ... x_{j_r} P_r} prod_{a=0}^r tau(P_a(V_u)),
-3.2 Wick testing.
+the sum over increasing factorization positions with empty gaps
+contributing one. Extend D_j linearly to all polynomials. Then there
+is an orthonormal family of semicircular Wick polynomials U_j(S) with
-Lemma 3.1 (Wick testing). Let C be a centered bounded self-adjoint
-n-tuple, S a standard semicircular family free from C, and Z = C +
-sqrt(u) S with u > 0. For a multi-index j = (j_1, ..., j_r) with
-r >= 1 and a monomial P in x_1, ..., x_n, define
+ D_j(P) = u^{-r/2} tau(U_j(S)^* P(V_u)). (2)
- D_j(P) = sum over factorizations P = P_0 x_{j_1} P_1 ... x_{j_r} P_r
- of prod_{a=0}^r tau(P_a(Z)),
+In particular the conjugate variables of V_u exist and equal
-selecting increasing positions and letting empty gaps contribute one.
-Extend D_j linearly to polynomials. Then there is an orthonormal family
-of semicircular Wick polynomials U_j(S) with
+ xi_i = u^{-1/2} E[S_i | W^*(V_u)].
- D_j(P) = u^{-r/2} tau(U_j(S)^* P(Z)). (3.1)
+Proof. Construction. From the word S_{j_1} ... S_{j_r}, sum over
+subsets of pairwise disjoint adjacent pairs in the original index list,
+deleting each chosen pair (j_a, j_{a+1}) with weight -delta_{j_a, j_{a+1}}.
+An empty subset yields the original word. The definition gives
+U_j^* = U_{(j_r, ..., j_1)}.
-In particular, the conjugate variables of Z exist and equal
-
- xi_i = u^{-1/2} E[ S_i | W^*(Z) ].
-
-Proof. Definition. Build U_j from the word S_{j_1} ... S_{j_r} by
-summing over subsets of pairwise disjoint adjacent pairs in the original
-index list, deleting each chosen pair (j_a, j_a + 1) with weight
--delta_{j_a, j_{a+1}}. The empty subset gives the original word. This
-gives U_j^* = U_{(j_r, ..., j_1)}.
-
-Expansion. Expand tau(U_j^* P(Z)) by noncrossing cumulants of
-Z = C + sqrt(u) S. Freeness and semicircularity force every block
-touching an initial S position to be a pair: covariance delta_{ij}
-against another initial S, sqrt(u) delta_{ij} against an S coming from P.
+Multilinear expansion. Expand tau(U_j^* P(V_u)) via noncrossing
+cumulants after multilinear expansion V_u = C + sqrt(u) S. Freeness
+and semicircularity force every block touching an initial S position
+to be a pair, with covariance delta_{jk} against another initial S and
+sqrt(u) delta_{jk} against a selected S in P.
Cancellation of deletions. Fix a choice of C or S at every P-position
-in the multilinear expansion, and a noncrossing partition pi on all
-positions. Let A(pi) be the set of initial-initial pair blocks with
-originally adjacent endpoints. Contributions correspond exactly to
-subsets F of A(pi) (delete pairs in F, expand the rest, restore), and
-restoring adjacent pairs preserves both noncrossingness and the
-covariance factors. Their net coefficient is
-
- sum_{F subset A(pi)} (-1)^{|F|} = (1 - 1)^{|A(pi)|}.
-
-If pi contains any initial-initial pair, an innermost one has adjacent
-endpoints (its gaps are all initial and must pair inside), so A(pi) is
-nonempty and pi cancels. This covers nested internal pairs and never
-requires deleting newly adjacent positions.
-
-Surviving partitions. All initial positions pair externally with
-positions of P, in reverse order. Since U_j^* has initial indices
-j_r, ..., j_1, the selected indices in P occur as j_1, ..., j_r.
-External pair arches prevent any remaining block from joining distinct
-gaps, so each gap's trace factors out independently of the choices of
-C or S within it. Each external pair contributes sqrt(u), proving
-(3.1).
-
-Orthonormality. Apply the identity with C = 0 and u = 1. A Wick word
-tests to zero against shorter words and to a Kronecker delta against a
-word of matching degree; every other term of a Wick polynomial has
-smaller degree. Conjugate symmetry covers the remaining cases.
-
-Conjugate variables. For r = 1, D_i(P) = (tau tensor tau)((partial_i P)(Z)),
-so (3.1) and the definition of conditional expectation give
-xi_i = u^{-1/2} E[S_i | W^*(Z)]. QED.
-
-3.3 Fisher deficit bound.
-
-Proposition 3.2 (Squared-cumulant Fisher bound). Let C be centered
-bounded self-adjoint with diagonal covariance tau(C_i C_j) = delta_ij c_i,
-c_i >= 0. Let S be standard semicircular free from C. Put V_u = C +
-sqrt(u) S and v_i(u) = u + c_i for u > 0. Then for every u > 0,
-
- sum_i 1 / v_i(u) <= Phi*(V_u) <= n / u. (3.2)
-
-If, in addition, sum_{r >= 2} u^{-r} ||kappa_{r+1}(C)||_2^2 < infinity,
+and a noncrossing partition pi on all positions. Let A(pi) be the
+initial-initial pair blocks whose endpoints were previously adjacent.
+For fixed choices, contributions correspond exactly to subsets
+F subset A(pi): delete pairs in F, expand the remaining word, restore.
+Restoring adjacent pairs preserves noncrossingness and covariance
+factors, so the total coefficient is
+
+ sum_{F subset A(pi)} (-1)^{|F|} = (1 - 1)^{|A(pi)|}.
+
+If pi has any initial-initial pair, its innermost one has adjacent
+endpoints - intervening positions are all initial, and each must pair
+inside to avoid crossing - so A(pi) is nonempty and pi cancels. This
+argument covers nested internal pairs and deletes nothing newly
+adjacent after each step.
+
+Surviving partitions. In each surviving partition, all initial
+positions pair externally with positions of P, in reverse order. The
+initial indices of U_j^* are j_r, ..., j_1, so the selected indices in
+P occur as j_1, ..., j_r. External arch pairs prevent any remaining
+block from connecting distinct gaps. Summing the unrestricted
+partitions and the C/S choices within each gap gives the trace of that
+gap, even when entries of C fail to be free. Each external pair
+contributes sqrt(u), proving (2).
+
+Orthonormality. With C = 0 and u = 1, a Wick word tests to zero against
+shorter words and to a Kronecker delta against a word of its own
+degree; every other term of a Wick polynomial has smaller degree.
+Conjugate symmetry covers the remaining cases.
+
+Conjugate variables. For r = 1, D_i(P) = (tau tensor tau)((partial_i P)(V_u)),
+so (2) together with the definition of conditional expectation gives
+xi_i = u^{-1/2} E[S_i | W^*(V_u)]. QED.
+
+Proposition 1 (Squared-cumulant Fisher bound). Let C be centered,
+bounded, self-adjoint, with diagonal covariance tau(C_i C_j) = delta_ij c_i
+for c_i >= 0. Let S be a standard semicircular n-tuple free from C,
+and put V_u = C + sqrt(u) S with v_i(u) = u + c_i for u > 0. Then
+
+ sum_i 1 / v_i(u) <= Phi*(V_u) <= n / u. (3)
+
+If additionally sum_{r >= 2} u^{-r} ||kappa_{r+1}(C)||_2^2 < infinity,
then
- 0 <= Phi*(V_u) - sum_i 1 / v_i(u)
- <= u^{-2} sum_{r >= 2} u^{-r} ||kappa_{r+1}(C)||_2^2. (3.3)
+ 0 <= Phi*(V_u) - sum_i 1 / v_i(u)
+ <= u^{-2} sum_{r >= 2} u^{-r} ||kappa_{r+1}(C)||_2^2. (4)
-Proof. General bounds. Set Z = V_u. Lemma 3.1 supplies xi_i, so
-projection contraction gives Phi*(Z) <= n / u. Testing the conjugate
-identity with Z_i gives tau(Z_i xi_i) = 1. Since ||Z_i||_2^2 = v_i(u),
+Proof. Bounds. Put Z = V_u. Lemma 3 supplies xi_i, and conditional
+expectation contraction gives Phi*(Z) <= n / u. Testing the conjugate
+identity with Z_i gives tau(Z_i xi_i) = 1; since ||Z_i||_2^2 = v_i(u),
Cauchy-Schwarz gives ||xi_i||_2^2 >= 1 / v_i(u), and summing proves the
-lower bound in (3.2).
+left half of (3).
-Deficit bound. Extract the block containing the first position in
-tau(Z_i P(Z)). Its other positions select letters in D_j(P), and its
-gaps contribute their moments:
+Cumulant expansion. Extract the block containing the first position in
+tau(Z_i P(Z)). Its other positions select letters in D_j(P); its gaps
+contribute moments:
- tau(Z_i P(Z)) = sum_{r >= 1, j}
- kappa_{r+1}(Z_i, Z_{j_1}, ..., Z_{j_r}) D_j(P).
+ tau(Z_i P(Z)) = sum_{r >= 1, j}
+ kappa_{r+1}(Z_i, Z_{j_1}, ..., Z_{j_r}) D_j(P).
-Only r <= deg P contributes. Centering removes the order-one cumulant;
-diagonal covariance makes the order-two term v_i(u) D_i(P). Adding free
-semicircular noise leaves all higher cumulants equal to those of C.
+Only r <= deg P contributes. Centering removes the order-one cumulant.
+Diagonal covariance makes the order-two term v_i(u) D_i(P). Adding
+free semicircular noise leaves all higher cumulants equal to those of C.
-Residual. Define the ambient L^2 series
+Residual. The ambient L^2 series
- R_i = sum_{r >= 2, j}
- u^{-r/2} kappa_{r+1}(C_i, C_{j_1}, ..., C_{j_r}) U_j(S)^*.
+ R_i = sum_{r >= 2, j}
+ u^{-r/2} kappa_{r+1}(C_i, C_{j_1}, ..., C_{j_r}) U_j(S)^*
-The adjoint Wick family is orthonormal and summability gives L^2
-convergence with
+converges by orthonormality of the adjoint Wick family, with
- sum_i ||R_i||_2^2 = sum_{r >= 2} u^{-r} ||kappa_{r+1}(C)||_2^2.
+ sum_i ||R_i||_2^2 = sum_{r >= 2} u^{-r} ||kappa_{r+1}(C)||_2^2.
-For fixed P, terms with r > deg P have zero trace pairing against P(Z)
-by (3.1). Since A -> tau(A P(Z)) is L^2 continuous, the truncation gives
+For fixed P, terms with r > deg P vanish in trace against P(V_u) by
+Lemma 3. Continuity of A -> tau(A P(V_u)) on L^2 and finite truncation
+of the cumulant expansion give
- tau( (Z_i - v_i(u) xi_i) P(Z) ) = tau(R_i P(Z)).
+ tau((Z_i - v_i(u) xi_i) P(V_u)) = tau(R_i P(V_u)).
-The trace pairing is bilinear and needs no complex conjugation. Testing
-with adjoints makes this Hilbert orthogonality; polynomial L^2 density
-identifies
+Trace is bilinear, so no complex conjugation is needed here. Testing
+with adjoints turns this into Hilbert-space orthogonality; polynomial
+L^2 density identifies
- Z_i - v_i(u) xi_i = E[R_i | W^*(Z)].
+ Z_i - v_i(u) xi_i = E[R_i | W^*(V_u)].
Assembly. Since xi_i is self-adjoint and tau(Z_i xi_i) = 1,
- ||Z_i - v_i(u) xi_i||_2^2 = v_i(u)^2 ( ||xi_i||_2^2 - 1/v_i(u) ).
-
-Summing after dividing by v_i(u)^2, and using projection contraction
-plus v_i(u) >= u, proves (3.3). QED.
-
-Remark. Two consequences are used later. The upper bound Phi*(V_u) <=
-n/u gives integrability of the Fisher-deficit integrand on every
-compact interval of positive noise times, so the free-flow identity
-(4.5) is legitimate. The deficit bound (3.3) reduces the large-noise
-entropy computation to control of ||kappa_{r+1}(C)||_2^2 for the signal
-C.
-
-
-==============================================================================
-4. THE SIGNAL AND THE TWO NOISE SCALES
-==============================================================================
-
-Requirements. Higher cumulants of the signal must be sparse (so the
-analytic bound holds), and many coordinates must share a block
-decomposition (so the matrix bound applies). Tensor independence
-supplies both at once.
-
-4.1 The signal.
-
-Fix p >= 2 and n = p + 1. Let B_1, ..., B_p be free variance-one
-semicirculars. In a separate tensor factor take Y uniform on p equally
-spaced points y_l = -1 + 2(l-1)/(p-1), l = 1, ..., p. Then Y commutes
-with every B_i but is not free from them. Set
-
- C = (B_1, ..., B_p, Y),
- V_a = C + sqrt(a) S,
- L(a) = (1/2) sum_{i=1}^n log(2 pi e (a + c_i)), c_i = tau(C_i^2),
-
-with S a standard semicircular n-tuple free from C. So c_i = 1 for
-i <= p, c_{p+1} = (1/p) sum_l y_l^2, and C is centered with diagonal
-covariance. L(a) is both the nonmicrostates entropy of the semicircular
-family with variances a + c_i and the classical entropy of the
-corresponding independent GUE law.
-
-4.2 Faithful realization.
-
-Choose uniformly bounded deterministic matrix models B^{(q_k)} of the
-free semicircular p-tuple, tensor with I_p, and use
-I_{q_k} tensor diag(y_1, ..., y_p) for Y. In dimensions d_k = p q_k
-these have the required tensor law. Adjoin independent GUE and take a
-subsequence satisfying moment tests with one fixed norm bound: the
-limiting moments give a positive tracial functional with asserted
-freeness, bounded on C_i by passing tr_d(Q^* D_i^2 Q) <= R_i^2
-tr_d(Q^* Q) to the limit. The GNS vacuum is cyclic for right
-multiplication (commuting with the weak closure of the left algebra)
-and separating there, so the state is faithful and normal. Faithfulness
-and semicircularity give ||B_i|| = ||S_i|| = 2.
-
-4.3 Cumulant bound.
-
-Lemma 4.1 (Cumulant bound). With K = 128 and C as above, for every
+ ||Z_i - v_i(u) xi_i||_2^2 = v_i(u)^2 ( ||xi_i||_2^2 - 1/v_i(u) ).
+
+Divide by v_i(u)^2, sum, and use projection contraction plus v_i(u) >= u
+to obtain (4). QED.
+
+Two consequences enter later. Equation (3) makes the Fisher deficit
+integrable on any compact interval of positive noise times, which is
+what makes the free-flow identity (8) legitimate; equation (4) reduces
+the large-noise entropy estimate to control of the squared higher-
+cumulant norms of the signal C.
+
+The witness signal. Fix a large integer p and set n = p + 1. Let
+B_1, ..., B_p be free variance-one semicirculars. In a separate tensor
+factor take Y uniform on p equally spaced points y_l = -1 + 2(l-1)/(p-1)
+for l = 1, ..., p, so Y commutes with every B_i but is not free from
+them. Put
+
+ C = (B_1, ..., B_p, Y),
+ V_a = C + sqrt(a) S,
+ L(a) = (1/2) sum_{i=1}^n log(2 pi e (a + c_i)), c_i = tau(C_i^2),
+
+where S is a standard semicircular n-tuple free from C. Then c_i = 1
+for 1 <= i <= p and c_n = (1/p) sum_l y_l^2, so C is centered with
+diagonal covariance. L(a) is at once the nonmicrostates entropy of the
+semicircular family with variances a + c_i and the normalized classical
+entropy of the corresponding independent GUE law.
+
+Faithful realization. Choose uniformly bounded deterministic matrix
+models B^{(q_k)} for the free semicircular p-tuple (Voiculescu's trace-
+convergence and norm-boundedness theorem), tensor with I_p, and use
+I_{q_k} tensor diag(y_1, ..., y_p) for Y; in dimensions d_k = p q_k
+these give models of the required tensor law. Adjoin an independent
+GUE n-tuple and select a subsequence satisfying successively more
+moment tests with a single fixed operator-norm bound. The joint
+limiting moments define a positive tracial functional on the variables
+(C, S) with the asserted freeness. In its GNS representation, left
+multiplication by C_i is bounded because the matrix inequality
+tr_d(Q^* D_i^2 Q) <= R_i^2 tr_d(Q^* Q) passes to the limit. The GNS
+vacuum is cyclic for right polynomial multiplication, which commutes
+with the weak closure of the left algebra, so it is separating for that
+closure and its vector state is faithful and normal. Faithfulness and
+the semicircular distribution give ||B_i|| = ||S_i|| = 2.
+
+Local PDF: ./p_nica-speicher.pdf
+Online: https://rolandspeicher.files.wordpress.com/2020/06/nica-speicher-book.pdf
+
+Lemma 4 (Cumulant bound). With K = 128 and C as above, for every
m >= 1,
- ||kappa_m(C)||_2 <= K^m p^{m/4}. (4.1)
+ ||kappa_m(C)||_2 <= K^m p^{m/4}. (5)
Proof. There are at most 4^m noncrossing partitions, with Mobius
-coefficients bounded by 4^m. Each product of block moments has modulus
-at most 2^m since ||B_i|| = 2 and ||Y|| <= 1. So each cumulant is at
+coefficients bounded by 4^m; each block moment has modulus at most 2^m
+because ||B_i|| = 2 and ||Y|| <= 1. So each individual cumulant is at
most 32^m.
-Suppose b of the m positions carry B-indices. Tensor independence
-factors each block moment into its B-word moment times its Y-moment. A
-nonzero Mobius sum requires, after deleting Y positions within each
-block, a semicircular word with a noncrossing pairing matching equal
-indices. Pairings from distinct blocks stay noncrossing (crossing
-them would cross the containing partition). So for fixed b positions at
-most 4^b p^{b/2} labelings contribute. Summing over positions bounds
-the potentially nonzero entries by (1 + 4 sqrt p)^m. Then
+Suppose b of the m positions carry B-indices. By tensor independence,
+each block moment factors into its B-word moment times its Y-moment.
+A nonzero Mobius sum requires, after deleting Y positions in each
+block, a semicircular word admitting a noncrossing pairing with equal
+indices; the union of these pairings is still noncrossing because
+crossings between distinct blocks would cross the containing
+partition. So for fixed b positions at most 4^b p^{b/2} labelings
+contribute. Summing over positions bounds the potentially nonzero
+entries by (1 + 4 sqrt p)^m. Then
- ||kappa_m(C)||_2 <= 32^m (1 + 4 sqrt p)^{m/2}
- <= (32 sqrt 5)^m p^{m/4}
- <= K^m p^{m/4}. QED.
+ ||kappa_m(C)||_2 <= 32^m (1 + 4 sqrt p)^{m/2}
+ <= (32 sqrt 5)^m p^{m/4}
+ <= K^m p^{m/4}. QED.
-4.4 Fisher deficit for the signal.
+Fisher deficit for the signal. Combine Proposition 1 with Lemma 4:
+the geometric series sum_{r >= 2} u^{-r} ||kappa_{r+1}(C)||_2^2 has
+ratio K^2 sqrt(p) / u, so for u > 2 K^2 sqrt(p) its value is at most
+8 K^6 p^{3/2} u^{-4} and
-Apply Proposition 3.2 with the cumulant bound (4.1). The series
-sum_{r >= 2} u^{-r} ||kappa_{r+1}(C)||_2^2 has ratio K^2 sqrt(p)/u;
-for u > 2 K^2 sqrt(p) it is < 1/2 and the geometric sum is bounded by
-2 K^6 p^{3/2} u^{-2}. Therefore
+ 0 <= Phi*(V_u) - sum_i 1 / (u + c_i) <= 2 K^6 p^{3/2} u^{-4}. (6)
- 0 <= Phi*(V_u) - sum_i 1/(u + c_i) <= 2 K^6 p^{3/2} u^{-4}. (4.2)
+Entropy deficit. Adding free semicircular noise of variance s to V_a
+yields V_{a+s}. By the definition of chi*,
-4.5 Entropy deficit formula.
+ chi*(V_a) = L(a) - (1/2) integral_a^infinity
+ [ Phi*(V_u) - sum_i 1 / (u + c_i) ] du. (7)
-Adding free semicircular noise of variance s to V_a gives V_{a+s}.
-Equation (2.2) then yields
+All terms are finite for a > 0: the integrand is measurable by Lemma 1,
+nonnegative by Proposition 1, and bounded by
- chi*(V_a) = L(a) - (1/2) integral_a^infinity
- [ Phi*(V_u) - sum_i 1/(u + c_i) ] du. (4.3)
+ sum_i c_i / (u (u + c_i)) <= sum_i c_i / u^2,
-All terms are finite for a > 0: the integrand is measurable (Lemma 2.1),
-nonnegative (Prop 3.2), and bounded by sum_i c_i / (u (u + c_i)) <=
-sum_i c_i / u^2, which is integrable on [a, infinity). The covariance
-correction is integral_0^infinity [(1+s)^{-1} - (a+c_i+s)^{-1}] ds =
-log(a + c_i), so no divergent integral is subtracted. In particular,
+which is integrable on [a, infinity). The covariance correction is
+integral_0^infinity [(1 + s)^{-1} - (a + c_i + s)^{-1}] ds = log(a + c_i),
+so no divergent integral is subtracted. In particular,
- chi*(V_a) >= L(a) - K^6 p^{3/2}/(3 a^3) > L(a) - 1/2
- whenever a > 2 K^2 sqrt(p), (4.4)
+ chi*(V_{h+t}) - chi*(V_h)
+ = (1/2) integral_0^t Phi*(V_{h+s}) ds, (8)
- chi*(V_{h+t}) - chi*(V_h)
- = (1/2) integral_0^t Phi*(V_{h+s}) ds. (4.5)
+ chi*(V_a) >= L(a) - K^6 p^{3/2} / (3 a^3) > L(a) - 1/2
+ whenever a > 2 K^2 sqrt(p). (9)
-4.6 Parameters.
+Parameters. Fix p large and t = p^{3/4}, so that
-Fix p large and set t = p^{3/4}, so that
+ t > 2 K^2 sqrt(p), 32 t / p < 1/8,
+ p / (256 (2 + t)) > 2 log(1 + 128 sqrt(2 + t)) + log 2 + 8. (10)
- t > 2 K^2 sqrt(p), 32 t / p < 1/8,
- p / (256 (2 + t)) > 2 log(1 + 128 sqrt(2 + t)) + log 2 + 8. (4.6)
+An explicit choice is p = 2^64 and t = 2^48: then 2 K^2 sqrt(p) = 2^47
+is less than t; 32 t / p = 2^{-11}; and 2 + t < 2^49 gives
+2^64 / (256 (2 + t)) > 2^64 / 2^57 = 128, while
+128 sqrt(2 + t) < 2^32 makes the right-hand side of the third
+inequality at most 67 log 2 + 8 < 55. From this point p, h, t are
+fixed; h will be chosen small next; only matrix dimensions tend to
+infinity.
-These are compatible because p/t = p^{1/4} grows faster than log p.
-Explicit choice: p = 2^64, t = 2^48. Then 2 K^2 sqrt(p) = 2^47 < t;
-32 t / p = 2^{-11}; and 2 + t < 2^49 makes the left side exceed
-2^64/2^57 = 2^7 = 128, beating 67 log 2 + 8 < 55 since
-128 sqrt(2 + t) < 2^32. From here on p, h, t are fixed; only matrix
-sizes d -> infinity.
+Spectral blocks. For h > 0 small, partition the spectrum of X_n = Y +
+sqrt(h) S_n at the midpoints of the y_l, and write e_l(h) for the
+resulting spectral projections. Let Delta = 2/(p - 1) be the spacing.
+Since ||sqrt(h) S_n|| = 2 sqrt(h), the condition h < Delta^2 / 64 places
+the spectrum inside the disjoint Delta/4-neighborhoods of the y_l,
+with every threshold in a spectral gap. Choose continuous functions
+agreeing with the interval indicators on those common neighborhoods.
+Continuous functional calculus, or uniform polynomial approximation on
+those neighborhoods followed by norm continuity, gives e_l(h) -> e_l(0)
+in norm. At h = 0 the e_l(0) commute with every B_i, have trace 1/p,
+and satisfy
-4.7 Spectral blocks.
+ sum_{i=1}^p sum_{l=1}^p ||e_l(0) B_i e_l(0)||_2^2
+ = sum_{i=1}^p ||B_i||_2^2 = p.
-For small h > 0, partition the spectrum of X_n = Y + sqrt(h) S_n at
-the midpoints between the y_l. Let e_l(h) be the resulting spectral
-projections and Delta = 2/(p-1) the atom spacing. Since
-||sqrt(h) S_n|| = 2 sqrt(h), the condition h < Delta^2 / 64 puts the
-spectrum in disjoint Delta/4-neighborhoods of the atoms, with every
-threshold in a spectral gap. Choose continuous functions agreeing
-with the interval indicators on those neighborhoods; continuous
-functional calculus gives e_l(h) -> e_l(0) in norm. At h = 0 the
-e_l(0) commute with every B_i, have trace 1/p, and satisfy
+There is therefore a fixed 0 < h < min{1, Delta^2/64} for which, with
+e_l = e_l(h) and X = V_h,
- sum_{i=1}^p sum_{l=1}^p ||e_l(0) B_i e_l(0)||_2^2
- = sum_{i=1}^p ||B_i||_2^2 = p.
+ tau(e_l) < 3 / (2 p) for each l,
+ sum_{i=1}^p sum_{l=1}^p ||e_l X_i e_l||_2^2 > 3 p / 4. (11)
-Therefore fix 0 < h < min{1, Delta^2/64} and write e_l = e_l(h). For
-X = V_h,
+Lemma 5 (Finite microstates entropy). The tuple X = V_h has
+chi(X) > -infinity.
- tau(e_l) < 3/(2 p) for each l,
- sum_{i=1}^p sum_{l=1}^p ||e_l X_i e_l||_2^2 > 3 p / 4. (4.7)
+Proof. Use deterministic models of C in dimensions d_k = p q_k and
+adjoin independent GUE noise of variance h. If M bounds the coordinate
+norms of the models, take R_0 = M + 8 sqrt(h). Lemma 2 and (1) place
+the noised models inside every fixed moment neighborhood of X at cutoff
+R_0 with probability tending to one. Their density is at most
+(d / (2 pi h))^{n d^2 / 2}. If q_d denotes that probability, then
-4.8 Finite microstates entropy.
-
-Lemma 4.2 (Finite chi). The tuple X = V_h has chi(X) > -infinity.
-
-Proof. Use the deterministic models for C in dimensions d_k = p q_k,
-then add independent GUE noise of variance h. If M bounds the
-coordinate norms of the models, take R_0 = M + 8 sqrt(h). Lemma 2.2
-and the GUE norm bound (2.3) show that these noised models lie in every
-fixed moment neighborhood of X at cutoff R_0 with probability tending
-to one. Their density is at most (d / (2 pi h))^{n d^2 / 2}. If q_d
-denotes that probability,
-
- Vol Gamma_{R_0}(X; m, d, epsilon)
- >= q_d (2 pi h / d)^{n d^2 / 2}.
+ Vol Gamma_{R_0}(X; m, d, epsilon) >= q_d (2 pi h / d)^{n d^2 / 2}.
Since q_d -> 1 along the model dimensions, the normalized limsup
-logarithm is at least (n/2) log(2 pi h). Take infimum over
-(m, epsilon), then supremum over cutoffs. The limsup definition allows
-restricting to this subsequence of dimensions. QED.
-
+logarithm is at least (n/2) log(2 pi h). Infimum over (m, epsilon) and
+supremum over R conclude. QED.
-==============================================================================
-5. ENTROPY PRODUCTION FOR SPECIFIED MATRIX ENSEMBLES
-==============================================================================
-
-Goal: transfer a terminal-time entropy bound back to the initial
-microstates entropy, via Jekel-Pi's classical-to-free Fisher
-comparison along the exact evolved ensemble.
-
-Lemma 5.1 (Entropy production). Fix X = V_h. Let R > 0 be a cutoff
-with chi_R(X) > -infinity. There exist dimensions d_k -> infinity and
+Lemma 6 (Entropy production). Fix X = V_h and a cutoff R > 0 with
+chi_R(X) > -infinity. There exist dimensions d_k -> infinity and
random tuples A^{(d)}, uniform on increasingly accurate positive-volume
-microstate sets of X at cutoff R, such that for independent standard
-GUE tuples G^{(d)},
+microstate sets at cutoff R, such that for independent standard GUE
+tuples G^{(d)},
- limsup_d H_d( A^{(d)} + sqrt(t) G^{(d)} )
- >= chi_R(X) + (1/2) integral_0^t Phi*(V_{h+s}) ds. (5.1)
+ limsup_d H_d( A^{(d)} + sqrt(t) G^{(d)} )
+ >= chi_R(X) + (1/2) integral_0^t Phi*(V_{h+s}) ds. (12)
-The limsup is over this particular ensemble.
+The limsup is over this specific ensemble.
-Proof.
-
-Entropy upper bound. Every uniform law on a positive-volume microstate
-set at cutoff R has sum_i E tr_d(A_i^2) <= n R^2. Comparison with
-independent GUE of variance R^2 bounds H_d(A) <= (n/2)log(2 pi e R^2).
-Thus chi_R(X) is a finite real number.
+Proof. Entropy upper bound. Every uniform law on a positive-volume
+microstate set at cutoff R has sum_i E tr_d(A_i^2) <= n R^2.
+Comparison with independent GUE of variance R^2 bounds H_d(A) from
+above by (n/2) log(2 pi e R^2); hence chi_R(X) is a finite real number.
Dimension sequence. Take d_k -> infinity with normalized log-volume at
-parameters (k, 1/k) at least chi_R(X) - 1/k. The pairs (k, 1/k) are
-cofinal among (m, epsilon); the sets have positive volume. So
+parameters (k, 1/k) at least chi_R(X) - 1/k. Such arbitrarily large
+dimensions exist by the defining limsup and the sets have positive
+volume. The pairs (k, 1/k) are cofinal among (m, epsilon). Their
+uniform laws satisfy
- liminf_d H_d(A^{(d)}) >= chi_R(X). (5.2)
+ liminf_d H_d(A^{(d)}) >= chi_R(X). (13)
Here and below d ranges over d_k. Every tuple in the support of A^{(d)}
-has fixed cutoff and successively accurate moments of X; the choice
-does not depend on the later noise time or polynomial tests.
+has the same fixed cutoff and increasingly accurate moments of X; the
+choice is independent of the later noise time or polynomial tests.
-De Bruijn identity. Put Z^{(d)}(s) = A^{(d)} + sqrt(s) G^{(d)} with
-density f_s for s > 0. In the n d^2 real Hilbert-Schmidt coordinates,
-the added Gaussian has covariance (s/d) I, so
-partial_s f_s = (1/(2 d)) Delta f_s. Differentiating entropy and
-integrating by parts:
+De Bruijn identity. Put Z^{(d)}(s) = A^{(d)} + sqrt(s) G^{(d)} and let
+f_s be its density for s > 0. In the n d^2 real Hilbert-Schmidt
+coordinates the added Gaussian has covariance (s/d) I, so
+partial_s f_s = (1/(2 d)) Delta f_s. Differentiating the entropy and
+integrating by parts gives
- d/ds H_d(Z^{(d)}(s)) = (1/2) I_d(s),
- I_d(s) = sum_i E tr_d(F_i^2) = d^{-3} integral |grad log f_s|^2 f_s,
- F_i = -(1/d) grad_i log f_s evaluated at Z^{(d)}(s). (5.3)
+ d/ds H_d(Z^{(d)}(s)) = (1/2) I_d(s),
+ I_d(s) = sum_i E tr_d(F_i^2) = d^{-3} integral |grad log f_s|^2 f_s,
+ F_i = -(1/d) grad_i log f_s evaluated at Z^{(d)}(s). (14)
Gaussian convolution of a compactly supported distribution justifies
both steps on compact positive-time intervals: derivatives have
-Gaussian decay, a Gaussian lower bound on f_s bounds |log f_s(x)| by
-C(1 + ||x||^2), and grad log f_s(x) = -(d/s)(x - E[A^{(d)} | Z^{(d)}(s)
-= x]) has at most linear growth. So entropy and Fisher information are
-finite at positive times.
+Gaussian decay, a Gaussian lower bound on f_s controls |log f_s(x)| by
+C (1 + ||x||^2), and grad log f_s(x) = -(d/s)(x - E[A^{(d)} | Z^{(d)}(s) = x])
+has at most linear growth. Entropy and Fisher information are
+therefore finite at positive times.
-Classical gain. Entropy is concave under mixtures of translations, so
-H_d(Z^{(d)}(epsilon)) >= H_d(A^{(d)}). Integrate epsilon -> t then let
-epsilon -> 0:
+Classical gain. Entropy is concave under mixtures of translations,
+so H_d(Z^{(d)}(epsilon)) >= H_d(A^{(d)}) for each epsilon > 0.
+Integrating from 0 to t and letting epsilon -> 0,
- H_d(Z^{(d)}(t)) >= H_d(A^{(d)}) + (1/2) integral_0^t I_d(s) ds. (5.4)
+ H_d(Z^{(d)}(t)) >= H_d(A^{(d)}) + (1/2) integral_0^t I_d(s) ds. (15)
No entropy continuity at zero is needed.
Matrix-to-free Fisher. For fixed s > 0 and any complex polynomial P,
- E tr_d(F_i P(Z^{(d)}(s)))
- = E (tr_d tensor tr_d)((partial_i P)(Z^{(d)}(s))). (5.5)
+ E tr_d(F_i P(Z^{(d)}(s)))
+ = E (tr_d tensor tr_d)((partial_i P)(Z^{(d)}(s))). (16)
-Normalization check. Fix a Hermitian basis E_alpha orthonormal for the
+Normalization. Fix a Hermitian basis E_alpha orthonormal for the
unnormalized Hilbert-Schmidt inner product; it satisfies
-sum_alpha E_alpha D E_alpha = Tr(D) 1. Scalar integration by parts
-(real and imaginary parts separately) turns the left side into
-d^{-2} sum_alpha E Tr(E_alpha D_i P(Z)[E_alpha]). At an occurrence
-P = L Z_i R in a monomial, its contribution is
-d^{-2} sum_alpha Tr(E_alpha L E_alpha R) = d^{-2} Tr(L) Tr(R) =
-tr_d(L) tr_d(R). Summing over occurrences proves (5.5) in the correct
-order.
-
-Complete squares. For self-adjoint P_i,
-
- I_d(s) >= sum_i E tr_d( 2 F_i P_i(Z^{(d)}(s)) - P_i(Z^{(d)}(s))^2 ).
-
-By (5.5) the score terms reduce to products of normalized polynomial
-traces. Lemma 2.2 then gives
-
- lim_d RHS = sum_i [ 2 (tau tensor tau)((partial_i P_i)(V_{h+s}))
- - tau(P_i(V_{h+s})^2) ].
-
-The limiting law is V_{h+s} because the fresh semicircular noise is free
-from X = V_h; conjugate variables exist by Lemma 3.1. Taking supremum
-over tests via Lemma 2.1:
-
- liminf_d I_d(s) >= Phi*(V_{h+s}). (5.6)
-
-Along the same d_k, for every fixed s > 0. Since s -> V_{h+s} has
-continuous moments, Phi*(V_{h+s}) is measurable (Lemma 2.1). Fatou on
-the nonnegative I_d, together with (5.2) and (5.4):
-
- liminf_d H_d(Z^{(d)}(t))
- >= chi_R(X) + (1/2) integral_0^t Phi*(V_{h+s}) ds.
+sum_alpha E_alpha D E_alpha = Tr(D) 1 for every complex D. Scalar
+integration by parts on real and imaginary parts turns the left side
+into
-This proves (5.1), with liminf in place of limsup. The free Fisher
-integral is finite since Phi*(V_u) <= n/u. Throughout the terminal law
-is the same A^{(d)} + sqrt(t) G^{(d)}; no entropy-maximizing law at the
-terminal distribution has been used. QED.
+ d^{-2} sum_alpha E Tr(E_alpha D_i P(Z)[E_alpha]),
+where D_i P(Z)[E] is the directional derivative in the i-th matrix
+coordinate along E. At an occurrence P = L Z_i R in a monomial its
+contribution is
-==============================================================================
-6. A PERSISTENT BLOCK-STRUCTURE ENTROPY DEFICIT
-==============================================================================
+ d^{-2} sum_alpha Tr(E_alpha L E_alpha R)
+ = d^{-2} Tr(L) Tr(R) = tr_d(L) tr_d(R).
-We bound the entropy of the terminal ensembles retained in Lemma 5.1.
-Fix its cutoff R, sequence d_k, ensembles A^{(d)}, and parameters p, h, t.
-For a decomposition of the identity into p orthogonal projections
-P = (P_1, ..., P_p), write D_P D = (sum_l P_l D_i P_l)_i and
-D_[p] = (D_1, ..., D_p).
+Summing over occurrences proves (16) with the prefix and suffix in the
+same order as the free difference quotient.
-6.1 Spectral witness.
-
-Take P as the midpoint partition of the last coordinate of A^{(d)}.
-Then for every tuple in the support of A^{(d)} and all large d,
-
- rank(P_l) <= 2 d / p, ||D_P A^{(d)}_[p]||_{2,d}^2 > p / 2. (6.1)
-
-Proof. Take a_d at cutoff R with moments converging to those of X.
-Let I_l be the midpoint intervals (any endpoint convention) and nu_d
-the empirical spectral measure of A_n^{(d)}. Weak convergence plus a
-union N of small closed neighborhoods of thresholds disjoint from
-sigma(X_n) give nu_d(N) -> 0. Pick continuous 0 <= f_l <= 1 agreeing
-with 1_{I_l} outside N. Then f_l(X_n) = e_l and F_l^{(d)} = f_l(A_n^{(d)})
-satisfies
+Complete squares. For self-adjoint P_i,
- ||P_l - F_l^{(d)}||_{2,d}^2 <= nu_d(N) -> 0.
+ I_d(s) >= sum_i E tr_d( 2 F_i P_i(Z^{(d)}(s)) - P_i(Z^{(d)}(s))^2 ).
-The cutoff gives ||P_l A_i^{(d)} P_l - F_l^{(d)} A_i^{(d)} F_l^{(d)}||_{2,d}
-<= 2 R ||P_l - F_l^{(d)}||_{2,d}; compressed norms are at most R, so
-squared norms also differ by o(1). Polynomial approximation and joint
-moment convergence yield
+By (16) the score terms become products of normalized polynomial
+traces. Lemma 2 then gives the limit of the right-hand side as
- rank(P_l)/d -> tau(e_l),
- ||P_l A_i^{(d)} P_l||_{2,d}^2 -> ||e_l X_i e_l||_2^2.
+ sum_i [ 2 (tau tensor tau)((partial_i P_i)(V_{h+s}))
+ - tau(P_i(V_{h+s})^2) ].
-Block summands are Hilbert-Schmidt orthogonal, so their norms add to
-||D_P A^{(d)}_[p]||^2. Strict margins in (4.7) then force (6.1).
-Uniformity over d follows by the same subsequence contradiction as in
-Lemma 2.2; closure points of microstate sets satisfy the same bounds.
+The limiting law is V_{h+s} because the fresh semicircular noise is
+free from X = V_h, and conjugate variables exist by Lemma 3. Taking
+the supremum over these tests via Lemma 1,
-6.2 The event.
+ liminf_d I_d(s) >= Phi*(V_{h+s}). (17)
-Put v = 1 + h + t. For a matrix tuple D let E_d be the set of D with
+This holds along the same d_k for every fixed s > 0. Continuous
+moments of s -> V_{h+s} make Phi*(V_{h+s}) measurable (Lemma 1).
+Fatou's lemma on the nonnegative I_d, then (13) and (15), yield
- ||D_[p]||_{2,d} <= 4 sqrt(v p), (6.2a)
- some P with rank(P_l) <= 2d/p
- and ||D_P D_[p]||_{2,d} >= sqrt(p)/4. (6.2b)
+ liminf_d H_d(Z^{(d)}(t))
+ >= chi_R(X) + (1/2) integral_0^t Phi*(V_{h+s}) ds.
-E_d is a deterministic Borel event: for each rank list that respects
-the rank bound, the space of decompositions is a compact unitary orbit;
-the max of ||D_P D_[p]|| over that orbit is Lipschitz in D because
-compressions are contractive. E_d does not depend on A^{(d)}; the
-spectral decomposition of A_n^{(d)} only witnesses its probability under
-the evolved law.
+That proves (12), even with liminf in place of limsup. QED.
-Lemma 6.1 (Event probability). Let mu_d be the law of
+Lemma 7 (Event probability). Let mu_d be the law of
A^{(d)} + sqrt(t) G^{(d)} and gamma_d the law of independent centered
-GUE coordinates of variances v_i(h+t) = h + t + c_i. Then for all
-large d,
-
- mu_d(E_d) >= 1/2, gamma_d(E_d) <= exp(-4 d^2). (6.3)
-
-Proof.
-
-Lower bound for mu_d. Fix A^{(d)} and its witness P; they are
-independent of G^{(d)}. Put r_l = rank(P_l). The block-diagonal
-Hermitian subspace has real dimension sum_l r_l^2. GUE covariance 1/d
-and the normalized trace give
-
- E[ ||sqrt(t) D_P G^{(d)}_[p]||_{2,d}^2 | A^{(d)} ]
- = t p sum_l (r_l / d)^2 <= 2 t.
+GUE coordinates of variances v_i(h+t) = h + t + c_i. For all
+sufficiently large d,
-Markov at threshold p/16 bounds failure by 32 t / p < 1/8. On success,
+ mu_d(E_d) >= 1/2, gamma_d(E_d) <= exp(-4 d^2). (20)
- ||D_P (A + sqrt(t) G)_[p]||_{2,d}
- >= ||D_P A_[p]||_{2,d} - ||D_P sqrt(t) G_[p]||_{2,d}
- > sqrt(p/2) - sqrt(p)/4 > sqrt(p)/4.
+Proof. Witness for mu_d. By (11) and Lemma 2, the empirical spectral
+decomposition of A_n^{(d)} supplies a decomposition P with
+rank(P_l) <= 2d/p and
-Independence and centering give
-E||A_[p] + sqrt(t) G_[p]||^2 = E||A_[p]||^2 + t p -> v p.
-Markov at 16 v p bounds failure of (6.2a) by 1/16 + o(1) < 1/8.
-Union bound: mu_d(E_d) >= 13/16 > 1/2.
+ ||D_P A^{(d)}_[p]||_{2,d}^2 > p/2. (18)
-Upper bound for gamma_d. Enumerate at most (d+1)^p rank lists.
-Choose an operator-norm delta-net of U(d) with delta = 1/(64 sqrt(v)).
-A maximal delta-separated subset has unitary centers and is a delta-net.
-Its ambient operator-norm balls of radius delta/2 are disjoint inside
-the ball of radius 1 + delta/2 in real 2 d^2-dimensional M_d(C); volume
-bounds the cardinality by (1 + 2/delta)^{2 d^2}.
+Fix A^{(d)} together with its witness P; both are independent of
+G^{(d)}. Put r_l = rank(P_l). GUE covariance 1/d and normalized
+trace give
-Each decomposition of a fixed rank list is a unitary conjugate of the
-standard coordinate blocks P^0 (including zero ranks). For
-||U - V|| <= delta,
+ E[ ||D_P sqrt(t) G^{(d)}_[p]||_{2,d}^2 | A^{(d)} ]
+ = t p sum_l (r_l/d)^2 <= t p * p * (2/p)^2 = 4 t.
- ||U^* D_[p] U - V^* D_[p] V||_{2,d} <= 2 delta ||D_[p]||_{2,d}.
+Markov at squared threshold p/16 bounds failure by 64 t/p < 1/4. On
+success, the triangle inequality and (18) yield
-Fixed-block compression is contractive; replacing a witness unitary by a
-net point loses at most 2 delta * 4 sqrt(vp) = sqrt(p)/8 in (6.2b). So
-some rank list and some net unitary U satisfy
+ ||D_P (A + sqrt(t) G)_[p]||_{2,d}
+ >= sqrt(p/2) - sqrt(p)/4 > sqrt(p)/4.
- ||D_{P^0}(U^* D_[p] U)||_{2,d} >= sqrt(p)/8.
+Centeredness and independence also give
-For fixed list and unitary, unitary invariance preserves the independent
-variance-v GUE law of the first p coordinates. The projected
-Hilbert-Schmidt vector has q = p sum_l rank(P_l^0)^2 <= 2 d^2 real
-coordinates of variance v/d. The normalized squared norm is (v/d^2) Q
-with Q chi-squared of degree q. Since E e^{Q/4} = 2^{q/2}, the
-probability that the squared norm reaches p/64, equivalently
-Q >= p d^2/(64 v), is at most
-
- exp( d^2 (log 2 - p/(256 v)) ). (6.4)
-
-The union bound has logarithm at most
-
- p log(d+1) + d^2 ( 2 log(1 + 128 sqrt(v)) + log 2 - p/(256 v) ).
+ E ||(A^{(d)} + sqrt(t) G^{(d)})_[p]||_{2,d}^2 = v p (1 + o(1)),
-Since v < 2 + t, (4.6) makes the bracket less than -8; the first term
-is o(d^2). So gamma_d(E_d) <= exp(-4 d^2) for large d. QED.
+which is at most 2 v p for large d. Markov at squared norm 16 v p
+bounds failure by 1/8 + o(1). Union bound: both conditions hold with
+probability at least 3/4 + 7/8 - 1 = 5/8 > 1/2. That proves
+mu_d(E_d) >= 1/2.
-6.3 Proof of Theorem 1.
+Unitary net. Enumerate at most (d+1)^p admissible rank lists. For
+each, take an operator-norm delta-net of the unitary group with
+delta = 1/(64 sqrt(v)). A maximal delta-separated subset of U(d) has
+unitary centers and is a delta-net; its ambient operator-norm balls of
+radius delta/2 are disjoint inside the ball of radius 1 + delta/2 of
+real (2 d^2)-dimensional matrix space, giving cardinality at most
+(1 + 2/delta)^{2 d^2}.
-Retain the same cutoff, dimension sequence, and ensembles. Put
-alpha = mu_d(E_d), beta = gamma_d(E_d). Binary data processing for
-relative entropy plus the binary-entropy bound give
+Local PDF: ./p_szarek.pdf
+Online: https://arxiv.org/pdf/math/9701213
- KL(mu_d || gamma_d) >= alpha log(alpha/beta)
- + (1-alpha) log((1-alpha)/(1-beta))
- >= alpha log(1/beta) - log 2.
+Compression and reduction. Every decomposition of a fixed rank list
+(including zero ranks) is a unitary conjugate of the standard
+coordinate blocks P^0. For unitaries U, V with ||U - V|| <= delta,
-Lemma 6.1 gives liminf_d d^{-2} KL(mu_d || gamma_d) >= 2.
+ ||U^* D_[p] U - V^* D_[p] V||_{2,d} <= 2 delta ||D_[p]||_{2,d}.
-Upper bound on H_d. At positive time the input is bounded, so the
-Gaussian convolution has finite entropy and finite second moments. The
-Gaussian reference density is
+Fixed-block compression is contractive, so replacing a witnessing
+unitary by a net point loses at most 2 delta * 4 sqrt(vp) = sqrt(p)/8
+in the event condition (19). Thus some admissible rank list and some
+net unitary U satisfy
- g_d(D) = prod_{i=1}^n ( (d / (2 pi v_i(h+t)))^{d^2/2}
- exp(-d^2 tr_d(D_i^2) / (2 v_i(h+t))) ).
+ ||D_{P^0}( U^* D_[p] U )||_{2,d} >= sqrt(p)/8,
-Using -int f log f = -int f log g_d - KL(f || g_d):
+equivalently squared norm at least p/64.
- H_d(Z^{(d)}(t)) = (1/2) sum_i [ log(2 pi v_i(h+t))
- + E tr_d(Z_i^{(d)}(t)^2)/v_i(h+t) ]
- - d^{-2} KL(mu_d || gamma_d).
-
-The expected squared coordinates equal those of A_i^{(d)} plus t, and
-converge to v_i(h+t). So
-
- limsup_d H_d(Z^{(d)}(t)) <= L(h+t) - 1. (6.5)
-
-Assembly. The same terminal law is in Lemma 5.1. Match the lower
-bound (Lemma 5.1) against the upper bound (6.5):
+Chi-squared tail. For fixed list and unitary, unitary invariance
+preserves the independent variance-v GUE law on the first p coordinates.
+The projected Hilbert-Schmidt vector has q = p sum_l rank(P_l^0)^2 real
+coordinates of variance v/d, where q <= 2 d^2 because the ranks sum to
+d and each is at most 2d/p. The normalized squared norm is (v/d^2) Q
+with Q chi-squared of degree q. Since E e^{Q/4} = 2^{q/2}, the
+probability that Q reaches p d^2/(64 v) - equivalently that the squared
+norm reaches p/64 - is at most
- chi_R(X) + (1/2) integral_0^t Phi*(V_{h+s}) ds
- <= limsup_d H_d(Z^{(d)}(t)) <= L(h+t) - 1.
+ exp( d^2 ( log 2 - p / (256 v) ) ).
-Rearranging,
+Union bound. The log of the full union bound is at most
- chi_R(X) <= L(h+t) - 1 - (1/2) integral_0^t Phi*(V_{h+s}) ds
- <= chi*(V_{h+t}) - 1/2 - (1/2) integral_0^t Phi* ds
- = chi*(V_h) - 1/2.
+ p log(d+1) + d^2 ( 2 log(1 + 128 sqrt(v)) + log 2 - p/(256v) ).
-The middle inequality uses (4.4) at a = h+t > 2 K^2 sqrt(p), which
-gives chi*(V_{h+t}) > L(h+t) - 1/2, hence L(h+t) - 1 < chi*(V_{h+t})
-- 1/2. The last equality is (4.5). The bound holds for every cutoff
-with chi_R(X) > -infinity; remaining cutoffs are automatic. Taking
-supremum over R gives chi(X) <= chi*(V_h) - 1/2. Lemma 4.2 shows
-chi(X) > -infinity; (4.3) at a = h shows chi*(V_h) finite. The
-bounded matrix realization in Section 4 supplies the required faithful
-normal tracial state and bounded self-adjoint tuple. QED.
+By v < 2 + t, (10) makes the bracket less than -8, and the first term
+is o(d^2) because p is fixed. So gamma_d(E_d) <= exp(-4 d^2) for
+large d. QED.
+Proof of the strict gap. Retain the same cutoff, dimension sequence,
+and ensembles throughout. Put alpha = mu_d(E_d) and beta = gamma_d(E_d).
+Binary data processing for relative entropy and the binary-entropy
+bound give
-==============================================================================
-REFERENCES
-==============================================================================
+ KL(mu_d || gamma_d) >= alpha log(alpha/beta)
+ + (1 - alpha) log((1 - alpha)/(1 - beta))
+ >= alpha log(1/beta) - log 2.
-[BCG] P. Biane, M. Capitaine, A. Guionnet. Large deviation bounds for
- matrix Brownian motion. Invent. Math. 152 (2003), 433-459.
+Lemma 7 yields liminf_d d^{-2} KL(mu_d || gamma_d) >= 2. Only one
+unit of this normalized deficit is retained in the comparison below.
-[Dab] Y. Dabrowski. A Laplace principle for Hermitian Brownian motion
- and free entropy I: the convex functional case. arXiv:1604.06420v2.
+Upper bound on H_d. At positive time the bounded-input Gaussian
+convolution has finite entropy and finite second moments, and its
+relative entropy with respect to gamma_d is finite. The Gaussian
+reference has density
-[Ge] L. Ge. Prime factors. Proc. Natl. Acad. Sci. USA 93 (1996),
- 12762-12763.
+ g_d(D) = prod_i ( (d / (2 pi v_i(h+t)))^{d^2/2}
+ exp( -d^2 tr_d(D_i^2) / (2 v_i(h+t)) ) ).
-[Gsurv] A. Guionnet. Large deviations and stochastic calculus for large
- random matrices. Probab. Surv. 1 (2004), 72-172.
+Using -int f log f = -int f log g_d - KL(f || g_d) and the
+normalization of H_d,
-[Jconv] D. Jekel. An elementary approach to free entropy theory for
- convex potentials. Anal. PDE 13 (2020), 2289-2374.
+ H_d(Z^{(d)}(t))
+ = (1/2) sum_i [ log(2 pi v_i(h+t)) + E tr_d(Z_i^{(d)}(t)^2)/v_i(h+t) ]
+ - d^{-2} KL(mu_d || gamma_d).
-[JP] D. Jekel, J. Pi. An elementary proof of the inequality
- chi <= chi* for conditional free entropy.
- Doc. Math. 29 (2024), 1085-1124.
+The expected squared coordinates equal those of A_i^{(d)} plus t,
+converging to v_i(h+t). Hence
-[NS] A. Nica, R. Speicher. Lectures on the Combinatorics of Free
- Probability. Cambridge University Press, 2006.
+ limsup_d H_d(Z^{(d)}(t)) <= L(h+t) - 1. (21)
-[FF] OpenAI. An isomorphism of the free group factors.
- OAI:An-isomorphism-of-the-free-group-factors-September-23-2026,
- 2026.
+Assembly. The same terminal ensemble appears in Lemma 6. Match the
+lower bound (12) against the upper bound (21):
-[Speicher] R. Speicher. Multiplicative functions on the lattice of
- non-crossing partitions and free convolution.
- Math. Ann. 298 (1994), 611-628.
+ chi_R(X) + (1/2) integral_0^t Phi*(V_{h+s}) ds
+ <= limsup_d H_d(Z^{(d)}(t)) <= L(h+t) - 1.
-[Szarek] S. J. Szarek. Metric entropy of homogeneous spaces.
- Banach Center Publ. 43 (1998), 395-410.
+Rearranging and applying (8) and (9) at a = h+t > 2 K^2 sqrt(p) gives
-[Vgue] D. Voiculescu. Limit laws for random matrices and free products.
- Invent. Math. 104 (1991), 201-220.
+ chi_R(X) <= L(h+t) - 1 - (1/2) integral_0^t Phi* ds
+ <= chi*(V_{h+t}) - 1/2 - (1/2) integral_0^t Phi* ds
+ = chi*(V_h) - 1/2,
-[Vchi] D. Voiculescu. The analogues of entropy and of Fisher's
- information measure in free probability theory. II.
- Invent. Math. 118 (1994), 411-440.
+for each cutoff with chi_R(X) > -infinity. For the remaining cutoffs
+the bound is automatic. Supremum over R gives
-[Vstar] D. Voiculescu. The analogues of entropy and of Fisher's
- information measure in free probability theory. V. Noncommutative
- Hilbert transforms. Invent. Math. 132 (1998), 189-227.
+ chi(X) <= chi*(V_h) - 1/2.
-[Vsurvey] D. Voiculescu. Free entropy. Bull. London Math. Soc. 34
- (2002), 257-278.
+Lemma 5 shows chi(X) > -infinity. Equation (7) at a = h shows
+chi*(V_h) finite.
+Related context. The companion preprint proves that L(F_2) and L(F_3)
+are isomorphic and shows each of the four free entropy dimensions of
+L(F_2) attains every integer n >= 2 on suitable generating tuples;
+Ge's prime factor analysis addresses the separate multiplicative
+structure of II_1 factors. Both are independent of the strict gap
+obtained above.
-==============================================================================
-END OF PAPER
-==============================================================================
+Local PDF: ./p_freegroup.pdf
+Local PDF: ./p_ge.pdf
+Online: https://europepmc.org/api/getPdf?pmcid=PMC23993
+The bounded self-adjoint tuple X = V_h from the witness construction
+supplies the required state and tuple. QED.