Matrix optimization under random external fields
Amir Dembo, Stanford University
Consider the problem of maximizing the quadratic form <x,Wx> + <h,x> over unit norm n-dimensional vectors x, where W is a Wigner matrix which is independent of the Gaussian vector h whose entries are independent and identically distributed. Two recent studies of the large n asymptotic probability of deviations of such maximum from its typical value, take very different approaches. Fyodorov and Le Doussal (2014) use the replica method of statistical physics, whereas Dembo and Zeitouni (2015) rely instead on the mathematical theory of large deviations. I will describe the main points of the latter, using this specific example also to illustrate some of the differences, strength and weaknesses of each approach.