Spectral order statistics of Gaussian random matrices: large deviations for trapped fermions and associated phase transitions

TL;DR

This paper analyzes the full order statistics of fermions in a harmonic trap by computing the probability distribution of the $k$th smallest eigenvalue in Gaussian random matrices, revealing phase transitions and large deviation behaviors.

Contribution

It provides an explicit rate function for the order statistics of eigenvalues, connecting large deviations, phase transitions, and Coulomb gas models in random matrix theory.

Findings

Probability of the $k$th eigenvalue follows a large deviation principle with a specific rate function.

The rate function exhibits quadratic behavior with a weak logarithmic singularity.

Phase transitions are identified in the Coulomb gas representation related to eigenvalue statistics.

Abstract

We compute the full order statistics of a one-dimensional gas of fermions in a harmonic trap at zero temperature, including its large deviation tails. The problem amounts to computing the probability distribution of the $k$th smallest eigenvalue $\lambda_{(k)}$ of a large dimensional Gaussian random matrix. We find that this probability behaves for large $N$ as $\mathcal{P}[\lambda_{(k)}=x]\approx \exp\left(-\beta N^2 \psi(k/N,x)\right)$, where $\beta$ is the Dyson index of the ensemble. The rate function $\psi(c,x)$, computed explicitly as a function of $x$ in terms of the intensive label $c=k/N$, has a quadratic behavior modulated by a weak logarithmic singularity at its minimum. This is shown to be related to phase transitions in the associated Coulomb gas problem. The connection with statistics of extreme eigenvalues of random matrices is also elucidated.