Matrix Optimization Under Random External Fields

Abstract

We consider the quadratic optimization problem

$$\begin{aligned}F_n^{W,\mathbf{h}}:= \sup _{\mathbf{x}\in S^{n-1}} \left( \frac{1}{2} \mathbf{x}^T W \mathbf{x}+ \mathbf{h}^T \mathbf{x}\right) \!, \end{aligned}$$

with $W$ a (random) matrix and $\mathbf{h}$ a random external field. We study the probabilities of large deviation of $F_n^{W,\mathbf{h}}$ for $\mathbf{h}$ 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 Fyodorov and Doussal (J Stat Phys 154:466–490, 2014).