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.
Highlights
In the 1950ies, it was found that certain important real N -particle ensembles can be described by aW
In the three remaining sections, we give an account on the three research areas we consider most important in connection with orthogonal polynomial ensembles: random matrix theory, random growth models, and non-colliding random processes
The circular orthogonal ensemble (COE) is the unique distribution on the set of orthogonal symmetric (N × N )-matrices that is invariant under conjugation with any real orthogonal matrix
Summary
In the 1950ies, it was found that certain important real N -particle ensembles (that is, joint distributions of N real random objects) can be described by a. Spectra distributions of certain random matrices (and the closely related non-colliding Brownian motions) were the only known important models that admit a description as in (1.1). The present text is an attempt to explain the problems and questions of interest in a unifying manner, to present solutions that have been found, to give a flavor of the methods that have been used, and to provide useful guidelines to much of the relevant literature It is aimed at the non-expert, the newcomer to the field, with a profound background in probability theory, who seeks a non-technical introduction, heuristic explanations, and a survey. In the three remaining sections, we give an account on the three research areas we consider most important in connection with orthogonal polynomial ensembles: random matrix theory, random growth models, and non-colliding random processes.
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