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Smooth Multi-Trace Statistics of Classical Ensembles: Large $N$ Expansions, Cumulants, and Matrix Integrals

Benoît Collins, Manasa Nagatsu

TL;DR

The paper advances the theory of large-$N$ expansions by developing a unified multi-trace framework for smooth test functions across classical random matrix ensembles, including GUE/GOE/GSE and Haar-distributed groups. By leveraging polynomial approximation, Bernstein-type inequalities, and Weingarten calculus, it derives explicit $1/N$-type (and often $1/N^{2k}$) expansions for multi-trace statistics and shows higher-order cumulants vanish, establishing Central Limit Theorems with precise covariance structures. The results illuminate the fluctuation structure of multi-matrix ensembles, and they enable formal $1/N^2$-expansions for matrix integrals with smooth potentials, offering a smooth-regularization perspective on formal matrix models. Together, these contributions connect combinatorial genus expansions with functional calculus in matrix probability, broadening applicability to spectral statistics, free energy computations, and multi-matrix fluctuations.

Abstract

We consider expectations of the form $E [tr h_1(X_1^N)... tr h_r(X_r^N)]$, where $X_i^N$ are self-adjoint polynomials in various independent classical random matrices and $h_i$ are smooth test function and obtain a large $N$ expansion of these quantities, building on the framework of polynomial approximation and Bernstein-type inequalities recently developed by Chen, Garza-Vargas, Tropp, and van Handel. As applications of the above, we prove the higher-order asymptotic vanishing of cumulants for smooth linear statistics, establish a Central Limit Theorem, and demonstrate the existence of formal asymptotic expansions for the free energy and observables of matrix integrals with smooth potentials.

Smooth Multi-Trace Statistics of Classical Ensembles: Large $N$ Expansions, Cumulants, and Matrix Integrals

TL;DR

The paper advances the theory of large- expansions by developing a unified multi-trace framework for smooth test functions across classical random matrix ensembles, including GUE/GOE/GSE and Haar-distributed groups. By leveraging polynomial approximation, Bernstein-type inequalities, and Weingarten calculus, it derives explicit -type (and often ) expansions for multi-trace statistics and shows higher-order cumulants vanish, establishing Central Limit Theorems with precise covariance structures. The results illuminate the fluctuation structure of multi-matrix ensembles, and they enable formal -expansions for matrix integrals with smooth potentials, offering a smooth-regularization perspective on formal matrix models. Together, these contributions connect combinatorial genus expansions with functional calculus in matrix probability, broadening applicability to spectral statistics, free energy computations, and multi-matrix fluctuations.

Abstract

We consider expectations of the form , where are self-adjoint polynomials in various independent classical random matrices and are smooth test function and obtain a large expansion of these quantities, building on the framework of polynomial approximation and Bernstein-type inequalities recently developed by Chen, Garza-Vargas, Tropp, and van Handel. As applications of the above, we prove the higher-order asymptotic vanishing of cumulants for smooth linear statistics, establish a Central Limit Theorem, and demonstrate the existence of formal asymptotic expansions for the free energy and observables of matrix integrals with smooth potentials.
Paper Structure (15 sections, 18 theorems, 152 equations)

This paper contains 15 sections, 18 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and algebraic preliminaries
  4. Definitions and asymptotic freeness of random matrices
  5. Technical preliminary results.
  6. Asymptotic expansion for multi-trace Gaussian ensembles
  7. The multi-trace GUE case
  8. Asymptotic expansion for multi-trace GOE and GSE
  9. Asymptotic expansion for Haar distributed random variables on compact matrix groups
  10. The Haar Unitary case
  11. The Orthogonal and symplectic case
  12. Applications of the asymptotic expansions
  13. Higher-order asymptotic vanishing of cumulants
  14. Central limit theorems
  15. Matrix integrals

Key Result

Theorem 2.4

Let $\bm{G}^N=(G_1^N,\ldots,G_d^N)$ be independent $N\times N$ GUE matrices, and let $\bm{U}^N=(U_1^N,\ldots,U_d^N)$ be independent Haar-distributed random matrices in $\mathop{\mathrm{U}}\nolimits (N)$. Let $\bm{s}=(s_1,\ldots,s_d)$ and $\bm{u}=(u_1,\ldots,u_d)$ be a free semicircular family and a and for any non-commutative polynomial $P\in {\mathbb{C}} \langle x_1,...,x_d, x_1^*,...,x_d^* \ran

Theorems & Definitions (39)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: Asymptotic freeness for GUE and Haar unitaries
  • Lemma 2.5: Bernstein inequality, Lemma 2.1 in VH_newapproach2
  • Lemma 2.6: Interpolation from $\tfrac{1}{N}$ samples, Proposition 3.1 in VH_newapproach2
  • Lemma 2.7: Rational Bernstein inequality, Lemma 7.5 in VH_newapproach2
  • Lemma 2.8: Extension of Bounded Multi-linear Maps
  • proof
  • Lemma 3.1
  • ...and 29 more