Journal of Statistical Physics

, Volume 159, Issue 6, pp 1306–1326 | Cite as

Matrix Optimization Under Random External Fields



We consider the quadratic optimization problem
$$\begin{aligned}F_n^{W,\mathbf{h}}:= \sup _{\mathbf{x}\in S^{n-1}} \left( \frac{1}{2} \mathbf{x}^T W \mathbf{x}+ \mathbf{h}^T \mathbf{x}\right) \!, \end{aligned}$$
with \(W\) a (random) matrix and \(\mathbf{h}\) a random external field. We study the probabilities of large deviation of \(F_n^{W,\mathbf{h}}\) for \(\mathbf{h}\) 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).


Large deviations Replica method Random matrices Spin glass 

Mathematics Subject Classification

60F10 82D30 



Amir Dembo: research partially supported by NSF Grant DMS-1106627. Ofer Zeitouni: research partially supported by a Grant from the Israel Science Foundation.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Stanford UniversityStanfordUSA
  2. 2.Weizmann Institute of ScienceRehovotIsrael
  3. 3.Courant InstituteNew YorkUSA

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