Mathematical Physics, Analysis and Geometry

, Volume 3, Issue 4, pp 339–373

Universality in Orthogonal and Symplectic Invariant Matrix Models with Quartic Potential

  • Alexandre Stojanovic
Article

DOI: 10.1023/A:1011457714198

Cite this article as:
Stojanovic, A. Mathematical Physics, Analysis and Geometry (2000) 3: 339. doi:10.1023/A:1011457714198

Abstract

In this work, we develop an orthogonal-polynomials approach for random matrices with orthogonal or symplectic invariant laws, called one-matrix models with polynomial potential in theoretical physics, which are a generalization of Gaussian random matrices. The representation of the correlation functions in these matrix models, via the technique of quaternion determinants, makes use of matrix kernels. We get new formulas for matrix kernels, generalizing the known formulas for Gaussian random matrices, which essentially express them in terms of the reproducing kernel of the theory of orthogonal polynomials. Finally, these formulas allow us to prove the universality of the local statistics of eigenvalues, both in the bulk and at the edge of the spectrum, for matrix models with two-band quartic potential by using the asymptotics given by Bleher and Its for the corresponding orthogonal polynomials.

random matriceseigenvaluescorrelation functionuniversality conjectureorthogonal polynomials

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Alexandre Stojanovic
    • 1
  1. 1.Institut de Mathématiques, Physique-mathématique et géométrieUniversité Paris 7Paris cedex 05France