Open Access
2005 Orthogonal polynomial ensembles in probability theory
Wolfgang König
Probab. Surveys 2: 385-447 (2005). DOI: 10.1214/154957805100000177


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.


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Wolfgang König. "Orthogonal polynomial ensembles in probability theory." Probab. Surveys 2 385 - 447, 2005.


Published: 2005
First available in Project Euclid: 30 November 2005

zbMATH: 1189.60024
MathSciNet: MR2203677
Digital Object Identifier: 10.1214/154957805100000177

Primary: 15A52 , 33C45 , 60-02 , 60C05 , 60F05 , 60K35 , 82C22 , 82C41
Secondary: 05E10 , 15A90 , 42C05

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

Rights: Copyright © 2005 The Institute of Mathematical Statistics and the Bernoulli Society

Vol.2 • 2005
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