Matrix Optimization Under Random External Fields

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