Journal of Statistical Physics

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

Matrix Optimization Under Random External Fields

  • Amir Dembo
  • Ofer Zeitouni


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.


  1. 1.
    Anderson, G.W., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices. Cambridge University Press, Cambridge (2010)MATHGoogle Scholar
  2. 2.
    Bai, Z.D., Yin, Y.Q.: Necessary and sufficient conditions for almost sure convergence of the largest eigenvalue of a Wigner matrix. Ann. Probab. 16, 1729–1741 (1988)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Ben Arous, G., Dembo, A., Guionnet, A.: Aging of spherical spin glasses. Probab. Theory Relat. Fields 120, 1–67 (2001)Google Scholar
  4. 4.
    Ben Arous, G., Guionnet, A.: Large deviations for Wigner’s law and Voiculescu’s non-commutative entropy. Probab. Theory Relat. Fields 108, 517–542 (1997)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, 2nd edn. Springer, New-York (1998)CrossRefMATHGoogle Scholar
  6. 6.
    Fyodorov, Y.V., Le Doussal, P.: Topology trivialization and large deviations for the minimum in the simplest random optimization. J. Stat. phys. 154, 466–490 (2014)CrossRefMATHMathSciNetGoogle Scholar

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

Personalised recommendations