Large Deviations of the Smallest Eigenvalue of the Wishart-Laguerre Ensemble

TL;DR

This paper investigates the large deviation behavior of the smallest eigenvalue in the Wishart-Laguerre ensemble, deriving rate functions for extreme fluctuations and confirming results with known cases.

Contribution

It introduces new universal rate functions for large fluctuations of the smallest eigenvalue, especially in nearly square matrices, using Coulomb gas techniques.

Findings

Derived rate functions for large deviations to the left and right of the hard edge.

Confirmed Coulomb gas predictions with exact results for β=1.

Identified universal rate functions for almost square matrices.

Abstract

We consider the large deviations of the smallest eigenvalue of the Wishart-Laguerre Ensemble. Using the Coulomb gas picture we obtain rate functions for the large fluctuations to the left and the right of the hard edge. Our findings are compared with known exact results for $\beta=1$ finding good agreement. We also consider the case of almost square matrices finding new universal rate functions describing large fluctuations.