Matrix optimization under random external fields

Abstract

We consider the quadratic optimization problem $$F_n^{W,h}:=\underset{x\in S^{n-1}}{\sup }(x^{T}Wx/2+h^{T}x),$$ 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).

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