Instantons and Extreme Value Statistics of Random Matrices

TL;DR

This paper explores the distribution of the largest eigenvalue in random Hermitian matrices, linking large deviation tails to instanton effects and string theory concepts, providing new computational and interpretative tools.

Contribution

It extends the orthogonal polynomials approach to calculate large deviations for the maximum eigenvalue, connecting tail behaviors to instanton corrections and string theory branes.

Findings

Left tail deviations follow perturbative large N expansion.

Right tail deviations relate to non-perturbative instanton effects.

Instanton action can be expressed via spectral curve.

Abstract

We discuss the distribution of the largest eigenvalue of a random N x N Hermitian matrix. Utilising results from the quantum gravity and string theory literature it is seen that the orthogonal polynomials approach, first introduced by Majumdar and Nadal, can be extended to calculate both the left and right tail large deviations of the maximum eigenvalue. This framework does not only provide computational advantages when considering the left and right tail large deviations for general potentials, as is done explicitly for the first multi-critical potential, but it also offers an interesting interpretation of the results. In particular, it is seen that the left tail large deviations follow from a standard perturbative large N expansion of the free energy, while the right tail large deviations are related to the non-perturbative expansion and thus to instanton corrections. Considering the standard interpretation of instantons as tunnelling of eigenvalues, we see that the right tail rate function can be identified with the instanton action which in turn can be given as a simple expression in terms of the spectral curve. From the string theory point of view these non-perturbative corrections correspond to branes and can be identified with FZZT branes.