Abstract: Consider the random variable X:=Tr(f1(W)A1...fk(W)Ak), where W is a Hermitian Wigner matrix, f1,...,fk are regular functions, and A1,...,Ak 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(fk+1(W)Ak+1...fk+ℓ(W)Ak+ℓ) of similar build. The results obtained match the structure of formulas in second-order free probability, which were previously only available for fj being polynomials.
Guillaume Barraquand et Raphaël Butez
