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Séminaire de Probabilités
# Multi-Point Functional CLT for Wigner Matrices

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Amphi Schwartz (IMT)
### Amphi Schwartz

#### IMT

Description

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.