A unified approach for large deviations of bulk and extreme eigenvalues of the Wishart ensemble

TL;DR

This paper introduces a unified Coulomb fluid-based method to analyze large deviations of both bulk and extreme eigenvalues in Wishart matrices, providing a comprehensive rate function and validating results with simulations.

Contribution

It develops a unified analytical framework to derive large deviation rate functions for bulk and extreme eigenvalues of Wishart matrices, connecting shifted index statistics with eigenvalue deviations.

Findings

Derived a rate function $ ext{Psi}(c, x)$ for large deviations

Connected bulk and edge eigenvalue deviations through limits

Validated analytical results with Monte Carlo simulations

Abstract

Within the framework of the Coulomb fluid picture, we present a unified approach to derive the large deviations of bulk and extreme eigenvalues of large Wishart matrices. By analysing the statistics of the shifted index number we are able to derive a rate function $\Psi (c, x)$ depending on two variables: the fraction $c$ of eigenvalues to the left of an infinite energetic barrier at position $x$. For a fixed value of $c$, the rate function gives the large deviations of the bulk eigenvalues. In particular, in the limits $c\to 0$ or $c\to 1$ it is possible to extract the left and right deviations of the smallest and largest eigenvalues, respectively. Alternatively, for a fixed value $x$ of the barrier, the rate function provides the large deviations of the shifted index number. All our analytical findings are compared with Metropolis Monte Carlo simulations, obtaining excellent agreement.