Multi-Point Functional Central Limit Theorem for Wigner Matrices

Abstract

Consider the random variable $\mathrm{Tr}(f_1(W)A_1\dots f_k(W)A_k)$ where $W$ is an $N\times N$ Hermitian Wigner matrix, $k\in \mathbb{N}$, and choose (possibly $N$-dependent) regular functions $f_1,\dots ,f_k$ as well as bounded deterministic matrices $A_1,\dots ,A_k$. We give a functional central limit theorem showing that the fluctuations around the expectation are Gaussian. Moreover, we determine the limiting covariance structure and give explicit error bounds in terms of the scaling of $f_1,\dots ,f_k$ and the number of traceless matrices among $A_1,\dots ,A_k$, thus extending the results of [Cipolloni, Erdős, Schröder 2023] to products of arbitrary length $k\ge 2$. As an application, we consider the fluctuation of $\mathrm{Tr}({\mathrm{e}}^{\mathrm{i}tW}{A}_1{\mathrm{e}}^{-\mathrm{i}tW}{A}_2)$ around its thermal value $\mathrm{Tr}(A_1)\mathrm{Tr}(A_2)$ when $t$ is large and give an explicit formula for the variance.

arXiv comment48 pages (including appendix)

References in corpus (13)

• On Asymptotic Properties of Large Random Matrices with Independent Entries
• Asymptotic expansion of beta matrix models in the one-cut regime
• Central Limit Theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices
• Random Matrices with Slow Correlation Decay
• Fluctuations of linear eigenvalue statistics of $\beta$ matrix models in the multi-cut regime
• Regularity conditions in the CLT for linear eigenvalue statistics of Wigner matrices
• Optimal local law and central limit theorem for $\beta$-ensembles
• Mesoscopic linear statistics of Wigner matrices of mixed symmetry class
• On Non-Gaussian Limiting Laws for the Certain Statistics of the Wigner Matrices
• Quantitative CLT for linear eigenvalue statistics of Wigner matrices
• Almost-optimal bulk regularity conditions in the CLT for Wigner matrices
• Mesoscopic eigenvalue statistics for Wigner-type matrices
• Fluctuation Moments for Regular Functions of Wigner Matrices