Large deviations of the shifted index number in the Gaussian ensemble

TL;DR

This paper derives a comprehensive rate function for large deviations in eigenvalue distributions of Gaussian matrices, covering bulk, extreme, and shifted index statistics, validated by simulations.

Contribution

It introduces a unified Coulomb fluid approach to derive a two-variable rate function capturing various large deviation phenomena in Gaussian eigenvalues.

Findings

Derived the rate function $,x)$ for eigenvalue deviations.

Validated analytical results with Monte Carlo simulations.

Connected eigenvalue statistics to trapped fermions in a harmonic potential.

Abstract

We show that, using the Coulomb fluid approach, we are able to derive a rate function $\Psi(c,x)$ of two variables that captures: (i) the large deviations of bulk eigenvalues; (ii) the large deviations of extreme eigenvalues (both left and right large deviations); (iii) the statistics of the fraction $c$ of eigenvalues to the left of a position $x$. Thus, $\Psi(c,x)$ explains the full order statistics of the eigenvalues of large random Gaussian matrices as well as the statistics of the shifted index number. All our analytical findings are thoroughly compared with Monte Carlo simulations, obtaining excellent agreement. A summary of preliminary results was already presented in [22] in the context of one-dimensional trapped spinless fermions in a harmonic potential.