Multi-Point Functional Central Limit Theorem for Wigner Matrices

Date
2024/02/09 Fri 15:30 - 17:00
Room
RIMS110号室
Speaker
Jana Reker
Affiliation
IST Austria
Abstract

Consider the random variable $X:=Tr( f_1(W)A_1 ... f_k(W)A_k)$, where $W$ is a Hermitian Wigner matrix, $f_1, ... , f_k$ are regular functions, and $A_1, ... , A_k$ are bounded deterministic matrices. In this talk, we study the fluctuations of $X$ around its expectation and give a functional central limit theorem on macroscopic and mesoscopic scales. Analyzing the underlying combinatorics further leads to explicit formulas for the variance of X as well as the covariance of $X$ and $Y:=Tr( f_{k+1}(W)A_{k+1} ... f_{k+\ell}(W)A_{k+\ell})$ of similar build. The results match the structure of formulas in second-order free probability, previously only available for $f_j$ being polynomials.