Central limit theorem for fluctuations of eigenvalues of real Wishart matrices

Abstract

We prove a central limit theorem for fluctuations of individual eigenvalues of real Wishart matrices, following the approach of Chhaibi-Sosoe (2022) for the Gaussian beta ensemble. Central limit theorems in random matrix theory have been studied for many types of statistics and models, but our understanding of fine-scale statistics on the level of individual eigenvalues is often limited by the precision of available tools. In this work, we pursue an approach that circumvents traditional difficulties. From the tridiagonal representation of random matrix models, we obtain a recurrence for the characteristic polynomial that is difficult to analyze directly. To address this issue, we make use of a transformation that considerably simplifies the recurrence. The transformation is inspired by the EFGP (Eggarter-Figotin-Gredeskul-Pastur) transformation in the study of random Schrödinger operators. The resulting process can be analyzed by a classical martingale argument, thus establishing the central limit theorem.