Consider the random variable Tr(f1β(W)A1ββ¦fkβ(W)Akβ) where
W is an NΓN Hermitian Wigner matrix, kβN, and choose
(possibly N-dependent) regular functions f1β,β¦,fkβ as well as bounded
deterministic matrices A1β,β¦,Akβ. 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 f1β,β¦,fkβ and the number of
traceless matrices among A1β,β¦,Akβ, thus extending the results of
[Cipolloni, Erd\H{o}s, Schr\"oder 2023] to products of arbitrary length
kβ₯2. As an application, we consider the fluctuation of
Tr(eitWA1βeβitWA2β)
around its thermal value Tr(A1β)Tr(A2β) when t is large
and give an explicit formula for the variance.Comment: 48 pages (including appendix