2023/07/20 by Jana Reker, Reker, Jana
Mathematics · #15B52 #60B20 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.2307.11028
openalex publication_date 2023/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider the random variable Tr( f1(W)A1… fk(W)Ak) where W is an N× N Hermitian Wigner matrix, k∈ℕ, and choose (possibly N-dependent) regular functions f1,…, fk as well as bounded deterministic matrices A1,…,Ak. 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 f1,…,fk and the number of traceless matrices among A1,…,Ak, thus extending the results of [Cipolloni, Erdős, Schröder 2023] to products of arbitrary length k≥2. As an application, we consider the fluctuation of Tr(ei tWA1e-i tWA2) around its thermal value Tr(A1)Tr(A2) when t is large and give an explicit formula for the variance.