Matrix Optimization Under Random External Fields

We consider the quadratic optimization problem F n W , h : = sup x ∈ S n - 1 1 2 x T W x + 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 Fyodorov and Doussal (J Stat Phys 154:466–490, 2014 ).