Matrix optimization under random external fields

TL;DR

This paper analyzes the large deviation probabilities of a quadratic optimization problem with random matrix and external field, validating and correcting predictions from the non-rigorous replica method.

Contribution

It provides rigorous probabilistic analysis of large deviations in quadratic optimization under random external fields, confirming and refining previous non-rigorous predictions.

Findings

Validated replica method predictions in certain regions.

Corrected replica method predictions in other regions.

Derived large deviation probabilities for different matrix ensembles.

Abstract

We consider the quadratic optimization problem $$F_n^{W,h}:= \sup_{x \in S^{n-1}} ( x^T W x/2 + h^T x )\,, $$ with $W$ a (random) matrix and $h$ a random external field. We study the probabilities of large deviation of $F_n^{W,h}$ for $h$ a centered Gaussian vector with i.i.d. entries, both conditioned on $W$ (a general Wigner matrix), and unconditioned when $W$ is a GOE matrix. Our results validate (in a certain region) and correct (in another region), the prediction obtained by the mathematically non-rigorous replica method in Y. V. Fyodorov, P. Le Doussal, J. Stat. phys. 154 (2014).