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Ultra high order cumulants and quantitative CLT for polynomials in Random Matrices

Zhigang Bao, Daniel Munoz George

TL;DR

The paper develops ultra high order cumulant bounds for traces of polynomials in self-adjoint random matrices from GUE/GOE and general Wigner ensembles, allowing deterministic matrices in the polynomials. By introducing quotient-graph combinatorics and genus-type expansions, the authors derive N-dependent bounds that remain meaningful even when r grows with N, revealing N^{2−r} scaling in Gaussian cases and N^{1−r/2} in general Wigner settings. These cumulant bounds fuel three quantitative CLTs for Tr P(X,D): a CLT with Cramér-type corrections, a Berry–Esseen bound, and a concentration inequality capturing Gaussian small-deviation tails and M-degree dependent large-deviation tails, where M is the polynomial degree. The results generalize second-order freeness by providing explicit rates and tail behavior for the full distribution of the trace, including polynomials in both random and deterministic matrices, with strong implications for fluctuations and concentration in non-invariant models. Overall, the work delivers a robust cumulant-based framework to obtain precise probabilistic laws for complex random-matrix polynomials in both Gaussian and non-Gaussian environments, with clear combinatorial and graphical underpinnings.

Abstract

From the study of the high order freeness of random matrices, it is known that the order $r$ cumulant of the trace of a polynomial of $N$-dimensional GUE/GOE is of order $N^{2-r}$ if $r$ is fixed. In this work, we extend the study along three directions. First, we also consider generally distributed Wigner matrices with subexponential entries. Second, we include the deterministic matrices into discussion and consider arbitrary polynomials in random matrices and deterministic matrices. Third, more importantly, we consider the ultra high order cumulants in the sense that $r$ is arbitrary, i.e., could be $N$ dependent. Our main results are the upper bounds of the ultra high order cumulants, for which not only the $N$-dependence but also the $r$-dependence become significant. These results are then used to derive three types of quantitative CLT for the trace of any given self-adjoint polynomial in these random matrix variables: a CLT with a Cramér type correction, a Berry-Esseen bound, and a concentration inequality which captures both the Gaussian tail in the small deviation regime and $M$-dependent tail in the large deviation regime, where $M$ is the degree of the polynomial. In contrast to the second order freeness which implies the CLT for linear eigenvalue statistics of polynomials in random matrices, our study on the ultra high order cumulants leads to the quantitative versions of the CLT.

Ultra high order cumulants and quantitative CLT for polynomials in Random Matrices

TL;DR

The paper develops ultra high order cumulant bounds for traces of polynomials in self-adjoint random matrices from GUE/GOE and general Wigner ensembles, allowing deterministic matrices in the polynomials. By introducing quotient-graph combinatorics and genus-type expansions, the authors derive N-dependent bounds that remain meaningful even when r grows with N, revealing N^{2−r} scaling in Gaussian cases and N^{1−r/2} in general Wigner settings. These cumulant bounds fuel three quantitative CLTs for Tr P(X,D): a CLT with Cramér-type corrections, a Berry–Esseen bound, and a concentration inequality capturing Gaussian small-deviation tails and M-degree dependent large-deviation tails, where M is the polynomial degree. The results generalize second-order freeness by providing explicit rates and tail behavior for the full distribution of the trace, including polynomials in both random and deterministic matrices, with strong implications for fluctuations and concentration in non-invariant models. Overall, the work delivers a robust cumulant-based framework to obtain precise probabilistic laws for complex random-matrix polynomials in both Gaussian and non-Gaussian environments, with clear combinatorial and graphical underpinnings.

Abstract

From the study of the high order freeness of random matrices, it is known that the order cumulant of the trace of a polynomial of -dimensional GUE/GOE is of order if is fixed. In this work, we extend the study along three directions. First, we also consider generally distributed Wigner matrices with subexponential entries. Second, we include the deterministic matrices into discussion and consider arbitrary polynomials in random matrices and deterministic matrices. Third, more importantly, we consider the ultra high order cumulants in the sense that is arbitrary, i.e., could be dependent. Our main results are the upper bounds of the ultra high order cumulants, for which not only the -dependence but also the -dependence become significant. These results are then used to derive three types of quantitative CLT for the trace of any given self-adjoint polynomial in these random matrix variables: a CLT with a Cramér type correction, a Berry-Esseen bound, and a concentration inequality which captures both the Gaussian tail in the small deviation regime and -dependent tail in the large deviation regime, where is the degree of the polynomial. In contrast to the second order freeness which implies the CLT for linear eigenvalue statistics of polynomials in random matrices, our study on the ultra high order cumulants leads to the quantitative versions of the CLT.

Paper Structure

This paper contains 20 sections, 20 theorems, 166 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Proof Strategy
  5. Organization
  6. Quotient graphs and partitions
  7. The set of partitions
  8. Oriented, unoriented and quotient graphs
  9. Sums associated with graphs
  10. Quotient graphs induced by graphs
  11. Expansion of the cumulants
  12. Upper bounds for $t(\cdot)$
  13. The Gaussian case
  14. The Wigner case
  15. Upper bounds for the cumulants
  16. ...and 5 more sections

Key Result

Theorem 1.4

Let $\mathcal{X}\sim$ GUE or GOE. Consider the polynomial $P(\mathcal{X},\mathcal{D})$ in (def of P) and the random variable $\mathbf{X}$ defined in (object). Suppose that $\mathbf{C}_1$ is satisfied. Then there exists a constant $\theta_1\equiv \theta_1 (K, M, t)\in (0, 1)$ such that for any $N$ and $r\geq 3$.

Figures (6)

  • Figure 1: Graph $G$
  • Figure 2: Unorientation of $G$
  • Figure 3: The quotient graph $\underline{G}^{\pi}$ of Example \ref{['Example: Quotient graph']}.
  • Figure 4: The graphs of Example \ref{['Example: order of a product of deterministic matrices']}.
  • Figure 5: The graphs of Example \ref{['Example: Graph induced by graphs']}.
  • ...and 1 more figures

Theorems & Definitions (59)

  • Example 1.1
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 49 more