Jana Reker: Multi-Point Functional CLT for Wigner Matrices

Abstract:

Consider the random variable \(X:=\operatorname{Tr}( f_1(W)A_1 ... f_k(W)A_k)\), where \(W\) is a Hermitian Wigner matrix, \(f_1, \dotsc, f_k\) are regular functions, and \(A_1, \dotsc, A_k\) 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 \(Y:= \operatorname{Tr}( f_{k+1}(W)A_{k+1} ... f_{k+\ell}(W)A_{k+\ell})\) of similar build. The results obtained match the structure of formulas in second-order free probability, which were previously only available for \(f_j\) being polynomials.