Probability Surveys

Orthogonal polynomial ensembles in probability theory

Wolfgang König

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Abstract

We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an orthogonal polynomial ensemble. The most prominent example is apparently the Hermite ensemble, the eigenvalue distribution of the Gaussian Unitary Ensemble (GUE), and other well-known ensembles known in random matrix theory like the Laguerre ensemble for the spectrum of Wishart matrices. In recent years, a number of further interesting models were found to lead to orthogonal polynomial ensembles, among which the corner growth model, directed last passage percolation, the PNG droplet, non-colliding random processes, the length of the longest increasing subsequence of a random permutation, and others.

Much attention has been paid to universal classes of asymptotic behaviors of these models in the limit of large particle numbers, in particular the spacings between the particles and the fluctuation behavior of the largest particle. Computer simulations suggest that the connections go even farther and also comprise the zeros of the Riemann zeta function. The existing proofs require a substantial technical machinery and heavy tools from various parts of mathematics, in particular complex analysis, combinatorics and variational analysis. Particularly in the last decade, a number of fine results have been achieved, but it is obvious that a comprehensive and thorough understanding of the matter is still lacking. Hence, it seems an appropriate time to provide a surveying text on this research area.

In the present text, we introduce various models, explain the questions and problems, and point out the relations between the models. Furthermore, we concisely outline some elements of the proofs of some of the most important results. This text is aimed at non-experts with strong background in probability who want to achieve a quick survey over the field.

Article information

Source
Probab. Surveys, Volume 2 (2005), 385-447.

Dates
First available in Project Euclid: 30 November 2005

Permanent link to this document
https://projecteuclid.org/euclid.ps/1133360670

Digital Object Identifier
doi:10.1214/154957805100000177

Mathematical Reviews number (MathSciNet)
MR2203677

Zentralblatt MATH identifier
1189.60024

Subjects
Primary: 15A52 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions] 60-02: Research exposition (monographs, survey articles) 60C05: Combinatorial probability 60F05: Central limit and other weak theorems 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C22: Interacting particle systems [See also 60K35] 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]
Secondary: 05E10: Combinatorial aspects of representation theory [See also 20C30] 15A90 42C05: Orthogonal functions and polynomials, general theory [See also 33C45, 33C50, 33D45]

Keywords
Random matrix theory Vandermonde determinant GUE orthogonal polynomial method bulk and edge scaling eigenvalue spacing Tracy-Widom distribution corner growth model noncolliding processes Ulam’s problem

Citation

König, Wolfgang. Orthogonal polynomial ensembles in probability theory. Probab. Surveys 2 (2005), 385--447. doi:10.1214/154957805100000177. https://projecteuclid.org/euclid.ps/1133360670


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