Correlation Functions, Cluster Functions, and Spacing Distributions for Random Matrices

1998/04/30 by Craig A. Tracy, Harold Widom · 14 citations
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Mathematical functions and polynomials #Random Matrices and Applications #math.SP #nlin.SI #solv-int

paper · pdf · doi:10.1023/a:1023084324803

published as J. Statistical Physics 92 (1998), 809-835. · 22 pages. LaTeX file. Minor correction

arxiv created 1998/06/27 · openalex publication_date 1998/09/01 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

The usual formulas for the correlation functions in orthogonal and symplectic matrix models express them as quaternion determinants. From this representation one can deduce formulas for spacing probabilities in terms of Fredholm determinants of matrix-valued kernels. The derivations of the various formulas are somewhat involved. In this article we present a direct approach which leads immediately to scalar kernels for unitary ensembles and matrix kernels for the orthogonal and symplectic ensembles, and the representations of the correlation functions, cluster functions and spacing distributions in terms of them.

Citations

• Sur la loi limite de l'espacement des valeurs propres d'une matrice ale´atoire

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• The largest sample eigenvalue distribution in the rank 1 quaternionic spiked model of Wishart ensemble
• A rate of convergence result for the largest eigenvalue of complex white Wishart matrices
• On Universality for Orthogonal Ensembles of Random Matrices
• Spacings: An Example for Universality in Random Matrix Theory
• Generalized Christoffel–Darboux formula for skew-orthogonal polynomials and random matrix theory
• Distribution of the largest eigenvalue for real Wishart and Gaussian random matrices and a simple approximation for the Tracy–Widom distribution
• Universality of the Local Spacing Distribution¶in Certain Ensembles of Hermitian Wigner Matrices
• Correlation Functions for β=1 Ensembles of Matrices of Odd Size
• Discrete Polynuclear Growth and Determinantal Processes
• Integrable Structure of Ginibre’s Ensemble of Real Random Matrices and a Pfaffian Integration Theorem