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2014

Matrix optimization under random external fields

Dembo, Amir, Zeitouni, Ofer

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

Matrix optimization under random external fields · Around

Built on

Z. D. Bai and Y. Q. Yin, Necessary and sufficient conditions for almost sure convergence of the largest eigenvalue of a Wigner matrix, Annals Probab. 16
1988
Earlier work this paper cites.
G. Ben Arous, A. Guionnet, Large deviations for Wigner’s law and Voiculescu’s non-commutative entropy. Probab. Theory Related Fields 108
1997
Earlier work this paper cites.

Similar

A. Dembo, O. Zeitouni, Large Deviations techniques and Applications , 2nd ed., Springer, New-York, 1998
1998
Cited alongside, same era.
G. Ben Arous, A. Dembo, A. Guionnet. Aging of spherical spin glasses. Probab. Theory Related Fields 120
2001
Cited alongside, same era.

Then

G. W. Anderson, A. Guionnet, O. Zeitouni, An Introduction to Random Matrices , Cambridge University press, Cambridge, 2010
2010
Later among the works it cites.
Y. V. Fyodorov, P. Le Doussal, Topology trivialization and large deviations for the minimum in the simplest random optimization. J. Stat. phys. 154
Original
2013
Later among the works it cites.

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