diff 45aa9fa..eeeea42 microstates_free_entropy_separation_v2.txt
diff --git a/microstates_free_entropy_separation_v2.txt b/microstates_free_entropy_separation_v2.txt
index a3fade7..8f68b1f 100644
--- a/microstates_free_entropy_separation_v2.txt
+++ b/microstates_free_entropy_separation_v2.txt
@@ -3,218 +3,288 @@
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 a faithful normal
+self-adjoint tuple X in a von Neumann algebra with faithful normal
tracial state such that
-infinity < chi(X) <= chi*(X) - 1/2 < infinity.
-Here chi is the original microstates entropy with an operator-norm
-cutoff and a limsup over matrix sizes. The counterexample uses a
-large but fixed number of variables.
-
+The construction uses a large but fixed number of variables.
1. INTRODUCTION
-Free entropy connects the joint moments of noncommuting variables with
-the geometry of their matrix approximations. For a bounded self-adjoint
-tuple X = (X_1, ..., X_n) in a tracial von Neumann algebra, Voiculescu's
-microstates free entropy chi(X) measures the asymptotic volumes of
-matrix tuples whose normalized trace moments approximate those of X.
-His nonmicrostates free entropy chi*(X) instead integrates free Fisher
-information along addition of freely independent semicircular noise.
-Both definitions, with normalizations, appear in Section 2.
+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.
-For a single self-adjoint variable these two quantities agree. Their
-agreement for general tuples was one of the questions in Voiculescu's
-unification program. The finite-entropy formulation asks whether
+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.
- chi(X) > -infinity implies chi(X) = chi*(X).
+For a single self-adjoint variable (n = 1), both invariants reduce to
+the same classical entropy.
-Guionnet stated this explicitly (Conjecture 7.1). The finiteness
-hypothesis is substantial: as discussed by Jekel and Pi, there exist
-nonapproximable laws with chi = -infinity < chi*. Our construction
-provides a strict gap with both entropies finite.
+1.2 The question.
-Main theorem. There exist an integer n >= 2 and a bounded self-adjoint
-tuple X = (X_1, ..., X_n) in a von Neumann algebra with faithful normal
+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
+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,
-
-where chi uses an operator-norm cutoff and a limsup over matrix sizes.
-
-This answers the finite-microstates-entropy equality question
-negatively. The number of variables is large but fixed; nothing is
-claimed for smaller prescribed n.
-
-Prior work. Biane, Capitaine, and Guionnet proved chi <= chi* through
-large-deviation bounds for matrix Brownian motion. Jekel and Pi gave an
-elementary proof using classical entropy and Fisher information, and
-extended it to conditional free entropy. Equality was known in
-strictly convex smooth-potential regimes (Dabrowski; Jekel). Those
-results also identify conditions under which the upper and lower
-matrix-size limits agree. Our example shows that finite entropy alone
-does not force equality.
-
-Idea of the proof. Two noise scales.
-
-Step 1 (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 let Y be uniformly distributed on p equally spaced points in
-[-1, 1]. Then Y commutes with every B_i but is not free from them.
-Write C = (B_1, ..., B_p, Y). For a standard semicircular n-tuple S
-free from C, put X = C + sqrt(h) S with h small. The tuple X has finite
-microstates entropy, because on the matrix side it is generated by
-deterministic models of C plus small Gaussian noise. Its last
-coordinate X_n = Y + sqrt(h) S_n partitions the spectrum into p blocks
-e_1, ..., e_p, and those blocks still contain most of the energy of the
-first p coordinates.
-
-Step 2 (matrix side). Uniform laws on increasingly accurate
-cutoff-microstate sets of X, evolved by independent GUE noise of
-variance t, retain a detectable simultaneous block decomposition: the
-probability is at least 1/2 along a suitable sequence of dimensions,
-whereas the corresponding independent GUE law with the same coordinate
-variances has probability at most exp(-4 d^2). Relative-entropy data
-processing then produces a normalized classical-entropy deficit of at
-least 1 in the limit.
-
-Step 3 (analytic side). For V_a = C + sqrt(a) S, Speicher's
-noncrossing-cumulant calculus and orthogonality of semicircular Wick
-polynomials give
-
- chi*(V_a) = L(a) - (1/2) integral_a^infinity Phi-deficit(u) du,
-
-where
+ -infinity < chi(X) <= chi*(X) - 1/2 < infinity.
- L(a) = (1/2) sum_{i=1}^n log(2 pi e (a + c_i)), c_i = tau(C_i^2),
+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.
-is the free entropy of the semicircular family with the same coordinate
-variances, and Phi-deficit(u) = Phi*(V_u) - sum_i 1 / (u + c_i). The
-integrand is bounded by squared higher-cumulant norms. The signal's
-cumulants are sparse, so at large a this deficit is small.
-
-Step 4 (transfer). The classical-to-free Fisher comparison of Jekel
-and Pi, applied to the exact terminal matrix ensemble, yields an
-entropy-production lower bound for that same terminal law in terms of
-the initial microstates entropy plus the free Fisher integral.
-
-Step 5 (matching). The terminal law's entropy is squeezed between the
-upper bound from Step 2 (deficit 1) and the lower bound from Step 4
-(deficit 0). This produces the gap.
-
-Two noise scales. The analytic deficit is of order t^{-3} p^{3/2},
-which is small whenever t >> sqrt(p). The Gaussian block estimate
-beats the unitary covering cost whenever t log(p) / p -> 0. The choice
-t = p^{3/4} satisfies both. We then choose one sufficiently large
-finite p, then a small finite h, and only then let matrix sizes tend to
-infinity with everything else fixed.
-
-Relation to companion work. A companion paper proves that 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
-within a fixed algebra; here we compare chi and chi* on a single tuple
-with both values finite.
-
-Organization. Section 2 fixes conventions and the matrix convergence
-facts. Section 3 proves the cumulant bound on the Fisher deficit.
-Section 4 constructs the signal, fixes both noise scales, and proves
-chi(X) > -infinity. Section 5 proves the specified-ensemble
-entropy-production lemma. Section 6 establishes the block event and
-combines everything into the proof of the main theorem.
==============================================================================
2. CONVENTIONS AND PRELIMINARY FACTS
==============================================================================
-Setting. All noncommutative variables live in a von Neumann algebra
-(M, tau) with a faithful normal tracial state tau. Write
+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||_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.
-for the Hilbert norm. A standard semicircular family has variance one
-in each coordinate and is free. Every noise family introduced below is
-free from the tuple it perturbs.
+2.2 Microstates entropy.
-Microstates. Let X = (X_1, ..., X_n) be a bounded self-adjoint tuple.
-For R > 0, an integer m >= 1, and epsilon > 0, let
+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
- Gamma_R(X; m, d, epsilon)
+ <A, B> = Re sum_{i=1}^n Tr(A_i^* B_i).
-be the set of self-adjoint d x d matrix tuples whose coordinate operator
-norms are at most R and whose normalized traces of every word of length
-at most m differ 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 Tr(A_i^* B_i). Set
+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_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).
- chi(X) = sup_R 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.
-The additive (n/2) log d normalizes like the entropy below. No-cutoff
-laws with chi = -infinity are allowed.
+2.3 Conjugate variables and free Fisher information.
-Conjugate variables. 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,
+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
+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 (PQ) = (partial_i P)(Q) + (P tensor 1)(partial_i Q),
+ partial_i x_j = delta_ij (1 tensor 1),
+ partial_i (P Q) = (partial_i P)(Q) + (P tensor 1)(partial_i Q),
-evaluated by substituting Z. Define Phi*(Z) = sum_i ||xi_i||_2^2 if
-they exist, and Phi*(Z) = infinity otherwise. Conjugate variables, when
-they exist, are unique and self-adjoint.
+applied by substituting Z. Define the free Fisher information
+
+ Phi*(Z) = sum_{i=1}^n ||xi_i||_2^2
+
+when the xi_i exist, and Phi*(Z) = infinity otherwise. Conjugate
+variables are unique and self-adjoint when they exist.
+
+2.4 A variational formula for Phi*.
Lemma 2.1 (Polynomial variational formula). If a bounded self-adjoint
-tuple Z has conjugate variables, then
+Z has conjugate variables, then
- Phi*(Z) = sup sum_{i=1}^n [ 2 (tau tensor tau)((partial_i P_i)(Z))
- - tau(P_i(Z)^2) ],
+ Phi*(Z) = sup sum_{i=1}^n ( 2 (tau tensor tau)((partial_i P_i)(Z))
+ - tau(P_i(Z)^2) ), (2.1)
the supremum being over self-adjoint noncommutative polynomials
-P_1, ..., P_n. It suffices to take P_i = (Q_i + Q_i^*)/2 with
-coefficients of Q_i in Q + i Q. Consequently, along a family of tuples
-with continuous moments and existing conjugate variables, Phi* is a
-measurable function of the parameter.
+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^*.
-Proof. 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 follows from polynomial L^2 density, so each xi_i
-is self-adjoint. The expression under the supremum equals
+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 ].
+ = 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.
-For the density: the closed span K of polynomial vectors is invariant
-under each bounded self-adjoint left multiplication operator Z_j. Its
-orthogonal projection commutes with Z_j and with W^*(Z); fixing 1, it
-fixes every bounded algebra vector, and those are dense in L^2. So
-K = L^2. Symmetrizing approximants gives the supremum over self-adjoint
-P_i; approximating coefficients by rationals gives a countable
-supremum. Each functional in that supremum is continuous in the
-moments, hence measurability of Phi*. QED.
+2.5 Nonmicrostates entropy.
-Nonmicrostates normalization. With S a standard semicircular family
-free from Z,
+Our 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.
+ [ n / (1 + s) - Phi*(Z + sqrt(s) S) ] ds. (2.2)
-The n / (1 + s) is the free Fisher contribution of n free semicircular
-coordinates; for a semicircular family the bracket vanishes identically.
+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).
-Matrix trace and GUE. Write tr_d = d^{-1} Tr and, for a matrix tuple D,
+2.6 Matrix notation and GUE.
+
+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).
@@ -227,66 +297,81 @@ classical entropy is
Independent GUE coordinates of variances w_i > 0 have
- H_d = (1/2) sum_{i=1}^n log(2 pi e w_i).
+ H_d = (1/2) sum_i log(2 pi e w_i).
This matches the microstates normalization exactly.
+2.7 Matrix convergence under bounded input.
+
+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
-normalized trace moments converge uniformly over K_d to the moments 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 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. Products of finitely many such traces converge in
-expectation. The dimensions may run over any sequence tending to
-infinity.
-
-Proof. Uniform norm bounds on GUE. 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|. For standard GUE,
-each v^* G v is centered real Gaussian of variance 1/d, since
-Tr((v v^*)^2) = 1. Hence
-
- P( ||G|| > L ) <= 2 exp( d (2 log 9 - L^2 / 8) ). (2.1)
-
-In particular the coordinate norms of any fixed finite GUE tuple are at
-most 8 with probability tending to one; integrating the tail yields
+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.
+
+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,
+
+ ||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
+
+ P( ||G|| > L ) <= 2 exp( d (2 log 9 - L^2 / 8) ). (2.3)
+
+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||.
-Gradient bound. Let a be deterministic with coordinate norms at most R
-and put F = tr_d P(a, G) for a fixed polynomial P. Fix 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 coordinates
-g_{j,alpha} are independent Gaussians of variance 1/d. Cyclicity of
-trace gives
+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.
+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),
where B_j is the sum of cyclically ordered products obtained from P by
-deleting each occurrence of the j-th GUE letter. For a polynomial in
-standard real Gaussian coordinates, expansion in orthonormal Hermite
-polynomials shows that variance is at most the expected squared
-gradient (each differentiation weights the squared modulus of a
-nonconstant coefficient by its positive total degree). Rescaling to
-variance 1/d, and applying separately to real and imaginary parts,
+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:
E |F - E F|^2 <= d^{-2} sum_j E tr_d(B_j^* B_j)
= O_{P,R}(d^{-2}).
-Since each B_j has fixed degree and a is uniformly bounded, the cited
-trace-convergence theorem (Voiculescu, arbitrary deterministic-tuple
-form) yields convergence in probability for every deterministic input
-sequence with the required moments.
+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. Suppose convergence over K_d failed: then for some
-polynomial and some positive threshold, a subsequence of dimensions and
-points a_d in K_d with error probability bounded below. These points
-form a bounded deterministic moment-convergent sequence, contradicting
-the previous conclusion. Conditioning on A^{(d)} gives convergence in
-probability for the random input.
+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)}.
Expectations. For a fixed product Q of normalized polynomial traces,
@@ -296,715 +381,637 @@ 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.
+
==============================================================================
3. A CUMULANT BOUND FOR THE FISHER DEFICIT
==============================================================================
-We compare Phi*(V_u) to the value sum_i 1 / v_i(u) that a semicircular
-family with the same covariance would have.
+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).
-Free cumulants. For a centered bounded self-adjoint n-tuple C, let
-kappa_m(C) be the order-m free cumulant array with
+3.1 Free cumulants.
+
+For a centered bounded self-adjoint n-tuple C, let kappa_m(C) be its
+order-m free cumulant array with
||kappa_m(C)||_2^2
= sum_{i_1, ..., i_m = 1}^n |kappa_m(C_{i_1}, ..., C_{i_m})|^2.
-We use the moment-cumulant formula and its Mobius inversion on
-noncrossing partitions, the vanishing of mixed cumulants between free
-algebras, and the fact that a standard semicircular family has only
-second cumulants equal to its covariance.
+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.
+
+3.2 Wick testing.
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
+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) = sum over factorizations P = P_0 x_{j_1} P_1 ... x_{j_r} P_r
+ 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)),
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) satisfying
+of semicircular Wick polynomials U_j(S) with
- D_j(P) = u^{-r/2} tau(U_j(S)^* P(Z)). (3.1)
+ D_j(P) = u^{-r/2} tau(U_j(S)^* P(Z)). (3.1)
In particular, the conjugate variables of Z exist and equal
xi_i = u^{-1/2} E[ S_i | W^*(Z) ].
-Proof. Definition. Define U_j from the word S_{j_1} ... S_{j_r} by
+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 yields the original word. This
-definition gives U_j^* = U_{(j_r, ..., j_1)}.
+-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 after
-multilinear expansion Z = C + sqrt(u) S. Freeness and semicircularity
-show that every block touching an initial S position is a pair; its
-covariance is delta_{ij} against another initial S and sqrt(u) delta_{ij}
-against a selected S in P.
+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.
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 its initial-initial pair blocks with originally
-adjacent endpoints. For these fixed choices, contributions correspond
-exactly to subsets F of A(pi): delete pairs in F, expand the remaining
-word, then restore. Restoring adjacent pairs preserves noncrossingness
-and the covariance factors, so their total coefficient is
+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 has an initial-initial pair, an innermost one has adjacent
-endpoints: its intervening positions are all initial and must each pair
-inside that interval to avoid crossing. Thus A(pi) is nonempty and the
-partition cancels. This includes nested internal pairs and does not use
-deletion of newly adjacent positions.
-
-Surviving partitions. In each surviving partition all initial positions
-pair externally with positions of P, in reverse order. The indices in
-U_j^* are j_r, ..., j_1, so the selected indices in P occur in order
-j_1, ..., j_r. External arches prevent any remaining block from
-connecting distinct gaps. Summing unrestricted partitions and choices
-of C or S in each gap gives the trace of the gap, even though entries of
-C need not be free. Each external pair contributes sqrt(u), proving
+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 a shorter word and produces a Kronecker delta
-against a word of its own degree. Every other term of a Wick polynomial
-has smaller degree, so U_j is orthonormal against Wick words of equal or
-smaller degree; conjugate symmetry covers the remaining cases.
+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.
-Proposition 3.2 (Squared-cumulant Fisher bound). Let C be a centered
-bounded self-adjoint n-tuple with diagonal covariance
-tau(C_i C_j) = delta_ij c_i, c_i >= 0. Let S be a standard semicircular
-family 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,
+3.3 Fisher deficit bound.
- sum_i 1 / v_i(u) <= Phi*(V_u) <= n / u. (3.2)
+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,
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)
+ <= u^{-2} sum_{r >= 2} u^{-r} ||kappa_{r+1}(C)||_2^2. (3.3)
-Proof. General bounds. Put Z = V_u. Conjugate variables xi_i come
-from Lemma 3.1, 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), Cauchy-Schwarz gives ||xi_i||_2^2 >= 1 / v_i(u),
-and summing proves the lower bound in (3.2).
+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),
+Cauchy-Schwarz gives ||xi_i||_2^2 >= 1 / v_i(u), and summing proves the
+lower bound in (3.2).
-Deficit bound. Start from 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 moments:
+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:
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,
-and diagonal covariance makes the order-two term v_i(u) D_i(P). Adding
-free semicircular noise leaves all higher cumulants unchanged.
+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
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 the summability assumption
-gives L^2 convergence with
+The adjoint Wick family is orthonormal and summability gives L^2
+convergence with
sum_i ||R_i||_2^2 = sum_{r >= 2} u^{-r} ||kappa_{r+1}(C)||_2^2.
-For a fixed P, terms with r > deg P have zero trace pairing with P(Z)
-by (3.1). Since A -> tau(A P(Z)) is continuous on L^2, the finite
-cumulant expansion gives
+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
- tau((Z_i - v_i(u) xi_i) P(Z)) = tau(R_i P(Z)).
+ tau( (Z_i - v_i(u) xi_i) P(Z) ) = tau(R_i P(Z)).
-The bilinear trace pairing needs no complex conjugation. Testing with
-adjoint polynomials converts this to Hilbert orthogonality, and
-polynomial L^2 density identifies
+The trace pairing is bilinear and needs no complex conjugation. Testing
+with adjoints makes this Hilbert orthogonality; polynomial L^2 density
+identifies
Z_i - v_i(u) xi_i = E[R_i | W^*(Z)].
-Assembly. Since xi_i are self-adjoint by their conditional-expectation
-formula and tau(Z_i xi_i) = 1,
+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) ).
- ||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.
-Summing after division by v_i(u)^2, using projection contraction and
-v_i(u) >= u, gives (3.3). QED.
==============================================================================
4. THE SIGNAL AND THE TWO NOISE SCALES
==============================================================================
-Competing requirements. The signal's higher cumulants must be sparse
-enough for the analytic bound, while many coordinates must share a
-block decomposition. Tensor independence supplies the latter without
-spoiling the former.
-
-Signal. Fix an integer p >= 2 and set n = p + 1. Let B_1, ..., B_p be
-free variance-one semicirculars. In a separate tensor factor take Y
-uniform on the p equally spaced points
-
- y_l = -1 + 2 (l - 1) / (p - 1), l = 1, ..., p.
-
-Thus 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)),
-
-where S is a standard semicircular n-tuple free from C and
-c_i = tau(C_i^2). Then c_i = 1 for i <= p, c_{p+1} = (1/p) sum_l y_l^2,
-and C is centered with diagonal covariance. The benchmark 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.
-
-Faithful realization and bounded matrix models. Choose uniformly
-bounded deterministic matrix models B^{(q_k)} of the free semicircular
-p-tuple (asymptotic freeness together with norm bounds from Section 2).
-Tensor with I_p and take 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 pass to a subsequence satisfying
-successively more moment tests and one fixed norm bound. The joint
-limiting moments define a positive tracial functional in (C, S) with
-the asserted freeness. In its GNS representation, left and right
-coordinate multiplication are bounded, because the matrix inequality
-
- tr_d(Q^* D_i^2 Q) <= R_i^2 tr_d(Q^* Q)
-
-passes to the limit. The vacuum is cyclic for right polynomial
-multiplication, which commutes with the weak closure of the left
-algebra, so it is separating for that closure. Its vector state is
-therefore faithful and normal; the trace identity extends by separate
-weak continuity. This supplies the required tracial von Neumann
-algebra. Further independent noise can be included. Faithfulness and
-the semicircular distribution give ||B_i|| = ||S_i|| = 2.
+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
m >= 1,
- ||kappa_m(C)||_2 <= K^m p^{m/4}. (4.1)
+ ||kappa_m(C)||_2 <= K^m p^{m/4}. (4.1)
-Proof. There are at most 4^m noncrossing partitions and their Mobius
-coefficients have magnitude at most 4^m (Catalan product formula,
-interval factorization of the noncrossing lattice). Each product of
-block moments has absolute value at most 2^m since ||B_i|| = 2 and
-||Y|| <= 1. Thus each individual cumulant is at most 32^m.
+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
+most 32^m.
-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 across blocks 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. Therefore
+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
- ||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.
-Consequence. Proposition 3.2 combined with a geometric series of ratio
-K^2 sqrt(p) / u < 1/2 for u > 2 K^2 sqrt(p) gives
+4.4 Fisher deficit for the signal.
- 0 <= Phi*(V_u) - sum_i 1 / (u + c_i)
- <= 2 K^6 p^{3/2} u^{-4}. (4.2)
+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
-Entropy deficit. Adding free semicircular noise of variance s to V_a
-produces V_{a+s}. By chi*'s definition,
+ 0 <= Phi*(V_u) - sum_i 1/(u + c_i) <= 2 K^6 p^{3/2} u^{-4}. (4.2)
- chi*(V_a) = L(a)
- - (1/2) integral_a^infinity
- [ Phi*(V_u) - sum_i 1 / (u + c_i) ] du. (4.3)
+4.5 Entropy deficit formula.
-All terms are finite for a > 0: the integrand is measurable
-(Lemma 2.1), nonnegative, and bounded above by
+Adding free semicircular noise of variance s to V_a gives V_{a+s}.
+Equation (2.2) then yields
- sum_i c_i / ( u (u + c_i) ) <= sum_i c_i / u^2,
+ chi*(V_a) = L(a) - (1/2) integral_a^infinity
+ [ Phi*(V_u) - sum_i 1/(u + c_i) ] du. (4.3)
-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 subtraction of divergent integrals is used. In particular,
+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,
- chi*(V_a) >= L(a) - K^6 p^{3/2} / (3 a^3) > L(a) - 1/2
- for a > 2 K^2 sqrt(p), (4.4)
+ 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. (4.5)
-
-Parameters. Fix p large enough 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. (4.6)
-
-These are compatible because p / t = p^{1/4} grows faster than log p.
-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 for the last inequality, since
-2 + t < 2^49 we have 2^64 / (256 (2 + t)) > 2^64 / 2^57 = 2^7 = 128,
-while 128 sqrt(2 + t) < 2^32 makes the right side less than
-67 log 2 + 8 < 55. From now on p, h, t are fixed and only matrix sizes
-tend to infinity.
-
-Spectral blocks. For small h > 0, partition the spectrum of
-X_n = Y + sqrt(h) S_n by midpoints between the atoms y_l; write
-e_l(h) for the resulting spectral projections and Delta = 2/(p - 1)
-for the atom spacing. Since ||sqrt(h) S_n|| = 2 sqrt(h), the condition
-h < Delta^2 / 64 places the spectrum inside disjoint Delta/4-neighborhoods
-of the atoms, with every threshold lying 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
+ = (1/2) integral_0^t Phi*(V_{h+s}) ds. (4.5)
+
+4.6 Parameters.
+
+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. (4.6)
+
+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.
+
+4.7 Spectral blocks.
+
+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
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.
-We may therefore fix 0 < h < min{1, Delta^2/64} and write e_l = e_l(h).
-For X = V_h,
+Therefore fix 0 < h < min{1, Delta^2/64} and write e_l = e_l(h). For
+X = V_h,
+
+ 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)
- 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)
+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 and
-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 (2.1) show
-that these noised models lie inside every fixed moment neighborhood of
-X at cutoff R_0, with probability tending to one. Their density is
-bounded above by (d / (2 pi h))^{n d^2 / 2}. If q_d denotes that
-probability,
+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}.
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). Take infimum over
+(m, epsilon), then supremum over cutoffs. The limsup definition allows
+restricting to this subsequence of dimensions. QED.
+
==============================================================================
5. ENTROPY PRODUCTION FOR SPECIFIED MATRIX ENSEMBLES
==============================================================================
-We transfer a later entropy bound for one evolved matrix law back to
-the initial microstates entropy. This uses the classical-to-free Fisher
-comparison of Jekel and Pi, keeping the terminal ensemble.
+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 are then 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)},
+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)},
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)
+ >= chi_R(X) + (1/2) integral_0^t Phi*(V_{h+s}) ds. (5.1)
-The limsup is over exactly this ensemble.
+The limsup is over this particular ensemble.
-Proof. Upper entropy 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 its normalized
-entropy above by (n/2) log(2 pi e R^2). Thus chi_R(X) is a finite real
-number.
+Proof.
-Choosing the dimension sequence. Take an increasing sequence d_k with
-the normalized log-volume at parameters (m, epsilon) = (k, 1/k) at
-least chi_R(X) - 1/k. Such arbitrarily large dimensions exist by the
-defining limsup, and the corresponding microstate sets have positive
-volume. The pairs (k, 1/k) are cofinal among all moment accuracies.
-The uniform laws satisfy
+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.
- liminf_d H_d(A^{(d)}) >= chi_R(X). (5.2)
+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
-Here and below, d ranges over the selected d_k. Every tuple in the
-support of A^{(d)} has the same fixed cutoff and increasingly accurate
-moments of X, and the selection does not depend on the subsequent noise
-time or polynomial tests.
+ liminf_d H_d(A^{(d)}) >= chi_R(X). (5.2)
-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. Thus
-partial_s f_s = (1/(2 d)) Laplacian f_s. Differentiating entropy and
-integrating by parts gives
+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.
- d/ds H_d(Z^{(d)}(s)) = (1/2) I_d(s), I_d(s) = sum_i E tr_d(F_i^2),
+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:
- F_i = -(1/d) grad_i log f_s evaluated at Z^{(d)}(s),
- I_d(s) = d^{-3} integral ||grad log f_s||^2 f_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). (5.3)
Gaussian convolution of a compactly supported distribution justifies
-differentiation and integration by parts 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. In particular the entropy and Fisher
-information are finite at positive times.
+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.
-Classical gain. Entropy concavity under mixtures of translations gives
-H_d(Z^{(d)}(epsilon)) >= H_d(A^{(d)}). Integrating from epsilon to t and
-letting epsilon -> 0 gives
+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:
- 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. (5.4)
-No continuity of entropy at zero is needed.
+No entropy continuity at zero is needed.
-Matrix-to-free Fisher. For fixed s > 0, integration by parts against
-any complex polynomial P gives
+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)
-
-To verify the normalization, take 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 for every complex D. Scalar
-integration by parts on real and imaginary parts turns the left side of
-(5.5) into
-
- d^{-2} sum_alpha E Tr( E_alpha D_i P(Z^{(d)}(s))[E_alpha] ),
-
-where D_i P(Z)[E] is the directional derivative in the ith matrix
-coordinate along E. At an occurrence P = L Z_i R in a monomial, its
-contribution is
+ = E (tr_d tensor tr_d)((partial_i P)(Z^{(d)}(s))). (5.5)
- d^{-2} sum_alpha Tr(E_alpha L E_alpha R)
- = d^{-2} Tr(L) Tr(R)
- = tr_d(L) tr_d(R).
+Normalization check. 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.
-Summing over occurrences proves (5.5) with the prefix and suffix in the
-same order as the free difference quotient.
+Complete squares. For self-adjoint P_i,
-Completing squares. For self-adjoint polynomial tests P_i,
+ I_d(s) >= sum_i E tr_d( 2 F_i P_i(Z^{(d)}(s)) - P_i(Z^{(d)}(s))^2 ).
- 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
-By (5.5) the score terms become products of normalized polynomial
-traces. Lemma 2.2 then gives the limit of the right side as
+ lim_d RHS = sum_i [ 2 (tau tensor tau)((partial_i P_i)(V_{h+s}))
+ - tau(P_i(V_{h+s})^2) ].
- 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:
-The limiting law is V_{h+s} because the new semicircular noise is free
-from X = V_h, and conjugate variables exist by Lemma 3.1. Taking
-supremum over tests via Lemma 2.1 gives
+ liminf_d I_d(s) >= Phi*(V_{h+s}). (5.6)
- liminf_d I_d(s) >= Phi*(V_{h+s}). (5.6)
-
-This holds for every fixed s > 0 along the same selected dimensions.
-The family s -> V_{h+s} has continuous moments, so Lemma 2.1 also gives
-measurability of Phi*(V_{h+s}). Fatou's lemma on the nonnegative
-functions I_d, then (5.2) and (5.4), yield
+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.
-This proves (5.1) with liminf in place of limsup. The integral is
-finite since Phi*(V_{h+s}) <= n/(h+s) and h > 0. Throughout, the
-terminal law is that of the same A^{(d)} + sqrt(t) G^{(d)}; no
-entropy-maximizing law at the terminal distribution has been chosen.
-QED.
+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.
+
==============================================================================
6. A PERSISTENT BLOCK-STRUCTURE ENTROPY DEFICIT
==============================================================================
We bound the entropy of the terminal ensembles retained in Lemma 5.1.
-Fix its cutoff R, dimension sequence, uniform microstate 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,
- D_{[p]} = ( D_1, ..., D_p ).
+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).
-Spectral witness. Use the midpoint spectral partition of the last
-coordinate of A^{(d)} to obtain P. Then for every tuple in the support
-of A^{(d)} and all sufficiently large dimensions,
+6.1 Spectral witness.
- rank(P_l) <= 2 d / p, || D_P A^{(d)}_{[p]} ||_{2,d}^2 > p / 2.
- (6.1)
+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,
-Uniformity of the witness. Suppose a deterministic sequence of tuples
-at cutoff R has joint moments converging to those of X. Let I_l be the
-midpoint intervals (with any fixed endpoint convention) and nu_d the
-empirical spectral measure of A_n^{(d)}. Since nu_d converges weakly to
-the law of X_n, choose a union N of small closed neighborhoods of the
-midpoint thresholds, disjoint from sigma(X_n). On a common compact
-interval containing [-R, R] and sigma(X_n), choose continuous functions
-0 <= f_l <= 1 agreeing with 1_{I_l} outside N. Then f_l(X_n) = e_l,
-and with F_l^{(d)} = f_l(A_n^{(d)}),
+ rank(P_l) <= 2 d / p, ||D_P A^{(d)}_[p]||_{2,d}^2 > p / 2. (6.1)
- || P_l - F_l^{(d)} ||_{2,d}^2 <= nu_d(N) -> 0.
+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
-Here weak convergence gives nu_d(N) -> 0 because N is closed with zero
-limiting mass. Small spectral outliers and eigenvalues exactly at
-thresholds are permitted.
+ ||P_l - F_l^{(d)}||_{2,d}^2 <= nu_d(N) -> 0.
-The cutoff bounds multiplication: for each i,
+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
- || 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}.
+ 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.
-Since the compressed norms are at most R, squared norms also differ by
-o(1). Uniform polynomial approximation of f_l, plus joint moment
-convergence, gives
+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.
- 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.
+6.2 The event.
-The block summands are orthogonal in Hilbert-Schmidt, so their squared
-norms add to || D_P A^{(d)}_{[p]} ||_{2,d}^2. The strict margins in
-(4.7) give (6.1). Uniformity over the selected supports holds; if it
-failed, a violating tuple in each dimension of a subsequence would form
-a bounded moment-convergent sequence, contradicting the argument.
-Closure points of the microstate sets satisfy the same moment bounds
-with weak inequalities whose errors tend to zero.
+Put v = 1 + h + t. For a matrix tuple D let E_d be the set of D with
-The event. Put v = 1 + h + t. For a matrix tuple D, let E_d be the
-set of D satisfying
+ ||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)
- ||D_{[p]}||_{2,d} <= 4 sqrt(v p),
-
- there is a decomposition P with rank(P_l) <= 2 d / p
- and || D_P D_{[p]} ||_{2,d} >= sqrt(p) / 4. (6.2)
-
-This is a deterministic Borel event. For each admissible rank list the
-space of projection decompositions is a compact unitary orbit, and the
-maximum of || D_P D_{[p]} ||_{2,d} on that orbit is continuous in D (in
-fact Lipschitz in the tuple 2-norm, since every compression is
-contractive). E_d is a finite union of closed threshold sets
-intersected with a closed norm ball. Its definition does not depend on
-A^{(d)}; the spectral decomposition of A_n^{(d)} only witnesses its
-probability under the evolved law.
+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.
Lemma 6.1 (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
-sufficiently large d,
+large d,
- mu_d(E_d) >= 1/2, gamma_d(E_d) <= exp(-4 d^2). (6.3)
+ mu_d(E_d) >= 1/2, gamma_d(E_d) <= exp(-4 d^2). (6.3)
Proof.
-Lower bound for mu_d. Use the witness of (6.1) before noise.
-Conditional on A^{(d)}, the projections P are fixed and independent of
-G^{(d)}. Put r_l = rank(P_l). The block-diagonal Hermitian subspace
-has real dimension sum_l r_l^2, so the 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.
+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
-Markov's inequality at threshold p/16 bounds failure by
-2 t / (p / 16) = 32 t / p < 1/8. On success, the projected total norm
-exceeds sqrt(p/2) - sqrt(p)/4 > sqrt(p)/4.
+ E[ ||sqrt(t) D_P G^{(d)}_[p]||_{2,d}^2 | A^{(d)} ]
+ = t p sum_l (r_l / d)^2 <= 2 t.
-Independence and centering also give
+Markov at threshold p/16 bounds failure by 32 t / p < 1/8. On success,
- E || A^{(d)}_{[p]} + sqrt(t) G^{(d)}_{[p]} ||_{2,d}^2
- = E || A^{(d)}_{[p]} ||_{2,d}^2 + t p
- -> v p.
+ ||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.
-Markov's inequality at 16 v p bounds the failure of the norm condition
-in (6.2) by 1/16 + o(1) < 1/8. The union bound proves
-mu_d(E_d) >= 3/4, stronger than claimed.
+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.
-Upper bound for gamma_d. Enumerate at most (d + 1)^p rank lists.
-Choose 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
-the real 2 d^2-dimensional space M_d(C); volume comparison bounds the
-cardinality by (1 + 2/delta)^{2 d^2}.
+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}.
-Every decomposition of a fixed rank list is a unitary conjugate of the
-standard coordinate blocks P^0 (including lists with zero ranks). For
-unitaries with || U - V || <= delta,
+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,
- || U^* D_{[p]} U - V^* D_{[p]} V ||_{2,d}
- <= 2 delta || D_{[p]} ||_{2,d}.
+ ||U^* D_[p] U - V^* D_[p] V||_{2,d} <= 2 delta ||D_[p]||_{2,d}.
-Fixed-block compression is contractive, so replacing a witnessing
-unitary by a net point loses at most 2 delta * 4 sqrt(v p) = sqrt(p)/8
-in (6.2). Thus some admissible rank list and some net unitary U
-satisfy
+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^0}( U^* D_{[p]} U ) ||_{2,d} >= sqrt(p)/8.
+ ||D_{P^0}(U^* D_[p] U)||_{2,d} >= sqrt(p)/8.
-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 <= 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
+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) ) ).
+ 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) ).
+ p log(d+1) + d^2 ( 2 log(1 + 128 sqrt(v)) + log 2 - p/(256 v) ).
-Since v < 2 + t, (4.6) makes the bracket less than -8. The first term
-is o(d^2) because p is fixed. This proves gamma_d(E_d) <= exp(-4 d^2)
-for large d. QED.
+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.
-Proof of the main theorem.
+6.3 Proof of Theorem 1.
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 and the bound on binary entropy give
+relative entropy plus the binary-entropy bound give
- KL(mu_d || gamma_d) >= alpha log(alpha / beta)
- + (1 - alpha) log((1 - alpha) / (1 - beta))
+ KL(mu_d || gamma_d) >= alpha log(alpha/beta)
+ + (1-alpha) log((1-alpha)/(1-beta))
>= alpha log(1/beta) - log 2.
-By Lemma 6.1,
+Lemma 6.1 gives liminf_d d^{-2} KL(mu_d || gamma_d) >= 2.
- liminf_d d^{-2} KL(mu_d || gamma_d) >= 2.
+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
-We retain one unit of this normalized deficit in the comparison.
+ 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))) ).
-At positive time the bounded-input Gaussian convolution has finite
-entropy and finite second moments, and its relative entropy to gamma_d
-is finite. The Gaussian reference has density
+Using -int f log f = -int f log g_d - KL(f || g_d):
- 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)) ) ).
+ 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).
-Using - integral f log f = - integral f log g_d - KL(f || g_d) together
-with the normalization of H_d,
+The expected squared coordinates equal those of A_i^{(d)} plus t, and
+converge to v_i(h+t). So
- 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).
+ limsup_d H_d(Z^{(d)}(t)) <= L(h+t) - 1. (6.5)
-The expected second moments equal those of A_i^{(d)} plus t, and
-converge to v_i(h+t). Hence
+Assembly. The same terminal law is in Lemma 5.1. Match the lower
+bound (Lemma 5.1) against the upper bound (6.5):
- limsup_d H_d(Z^{(d)}(t)) <= L(h+t) - 1. (6.4)
+ chi_R(X) + (1/2) integral_0^t Phi*(V_{h+s}) ds
+ <= limsup_d H_d(Z^{(d)}(t)) <= L(h+t) - 1.
-This concerns the exact terminal law Z^{(d)}(t) = A^{(d)} + sqrt(t) G^{(d)}
-of Lemma 5.1. Combining the two bounds on that entropy with (4.4) and
-(4.5):
+Rearranging,
- 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*(V_{h+s}) ds
- = chi*(V_h) - 1/2.
+ 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.
+
+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.
-This holds for every cutoff with chi_R(X) > -infinity; for the
-remaining cutoffs it is automatic. Taking the supremum over cutoffs
-preserves the inequality. Lemma 4.2 gives chi(X) > -infinity, and
-(4.3) with a = h gives chi*(X) finite. The bounded matrix realization
-in Section 4 supplies the required faithful normal tracial state and
-bounded self-adjoint tuple. This completes the proof. QED.
==============================================================================
REFERENCES
==============================================================================
-[BCG] P. Biane, M. Capitaine, and A. Guionnet,
- Large deviation bounds for matrix Brownian motion,
- Invent. Math. 152 (2003), no. 2, 433-459.
- doi:10.1007/s00222-002-0281-4.
-
-[Dab] Y. Dabrowski,
- A Laplace principle for Hermitian Brownian motion and free entropy I:
- the convex functional case,
- arXiv:1604.06420v2, October 31, 2017.
-
-[Ge] L. Ge,
- Prime factors,
- Proc. Natl. Acad. Sci. USA 93 (1996), no. 23, 12762-12763.
- doi:10.1073/pnas.93.23.12762.
-
-[Gsurv]
- A. Guionnet,
- Large deviations and stochastic calculus for large random matrices,
- Probab. Surv. 1 (2004), 72-172.
- doi:10.1214/154957804100000033.
-
-[Jconv]
- D. Jekel,
- An elementary approach to free entropy theory for convex potentials,
- Anal. PDE 13 (2020), no. 8, 2289-2374.
- doi:10.2140/apde.2020.13.2289.
-
-[JP] D. Jekel and J. Pi,
- An elementary proof of the inequality chi <= chi* for conditional
- free entropy,
- Doc. Math. 29 (2024), no. 5, 1085-1124.
- doi:10.4171/DM/969.
-
-[NS] A. Nica and R. Speicher,
- Lectures on the Combinatorics of Free Probability,
- London Mathematical Society Lecture Note Series, vol. 335,
- Cambridge University Press, 2006.
- doi:10.1017/CBO9780511735127.
-
-[FreeGroupFactors]
- OpenAI,
- An isomorphism of the free group factors,
- OpenAI Math Release preprint,
- OAI:An-isomorphism-of-the-free-group-factors-September-23-2026,
- 2026.
-
-[Speicher]
- R. Speicher,
- Multiplicative functions on the lattice of non-crossing
- partitions and free convolution,
- Math. Ann. 298 (1994), 611-628.
- doi:10.1007/BF01459754.
-
-[Szarek]
- S. J. Szarek,
- Metric entropy of homogeneous spaces,
- Banach Center Publ. 43 (1998), 395-410.
- arXiv:math/9701213.
-
-[Vgue] D. Voiculescu,
- Limit laws for random matrices and free products,
+[BCG] P. Biane, M. Capitaine, A. Guionnet. Large deviation bounds for
+ matrix Brownian motion. Invent. Math. 152 (2003), 433-459.
+
+[Dab] Y. Dabrowski. A Laplace principle for Hermitian Brownian motion
+ and free entropy I: the convex functional case. arXiv:1604.06420v2.
+
+[Ge] L. Ge. Prime factors. Proc. Natl. Acad. Sci. USA 93 (1996),
+ 12762-12763.
+
+[Gsurv] A. Guionnet. Large deviations and stochastic calculus for large
+ random matrices. Probab. Surv. 1 (2004), 72-172.
+
+[Jconv] D. Jekel. An elementary approach to free entropy theory for
+ convex potentials. Anal. PDE 13 (2020), 2289-2374.
+
+[JP] D. Jekel, J. Pi. An elementary proof of the inequality
+ chi <= chi* for conditional free entropy.
+ Doc. Math. 29 (2024), 1085-1124.
+
+[NS] A. Nica, R. Speicher. Lectures on the Combinatorics of Free
+ Probability. Cambridge University Press, 2006.
+
+[FF] OpenAI. An isomorphism of the free group factors.
+ OAI:An-isomorphism-of-the-free-group-factors-September-23-2026,
+ 2026.
+
+[Speicher] R. Speicher. Multiplicative functions on the lattice of
+ non-crossing partitions and free convolution.
+ Math. Ann. 298 (1994), 611-628.
+
+[Szarek] S. J. Szarek. Metric entropy of homogeneous spaces.
+ Banach Center Publ. 43 (1998), 395-410.
+
+[Vgue] D. Voiculescu. Limit laws for random matrices and free products.
Invent. Math. 104 (1991), 201-220.
- doi:10.1007/BF01245072.
-[Vchi] D. Voiculescu,
- The analogues of entropy and of Fisher's information measure
- in free probability theory. II,
+[Vchi] D. Voiculescu. The analogues of entropy and of Fisher's
+ information measure in free probability theory. II.
Invent. Math. 118 (1994), 411-440.
- doi:10.1007/BF01231539.
-
-[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.
- doi:10.1007/s002220050222.
-
-[Vsurvey]
- D. Voiculescu,
- Free entropy,
- Bull. London Math. Soc. 34 (2002), no. 3, 257-278.
- doi:10.1112/S0024609301008992.
+
+[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.
+
+[Vsurvey] D. Voiculescu. Free entropy. Bull. London Math. Soc. 34
+ (2002), 257-278.
==============================================================================