Matrix Optimization Under Random External Fields

Abstract

We consider the quadratic optimization problem (Formula presented.),with W a (random) matrix h a random external field. We study the probabilities of large deviation of (Formula presented.) for h a centered Gaussian vector with i.i.d. entries, both conditioned on $$W$$W (a general Wigner matrix), and unconditioned when $$W$$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).