Orthogonal polynomial ensembles in probability theory

By W. Koenig. 2004

187 citations recorded by the source corpus.

We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an {it 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.

Subfields: Applied Mathematics, Discrete Mathematics and Combinatorics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/4e7b11737eb0e182d429dfc4dc7ec619aca55af4

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In Applied Mathematics

In Discrete Mathematics and Combinatorics

In Statistics and Probability

Citations in the corpus

Conditional limit theorems for ordered random walks (2009, background): “A thorough review of these results can be found in [12].”
Random walks in cones (2011, background): “For two further Weyl chambers, WC := {x∈Rd : 0 x1 x2 · · · xd} and WD := {x ∈Rd : |x1| x2 · · · xd} and random walks with independent coordinates König and Schmid [33] have proven versions of Theorems 1 and 3 under moment conditions which are weaker than E|X|p ∞.”
Analysis of continuous strict local martingales via h-transforms (2007, background): “Please see the survey article by König [24] and the original paper by Dyson [7] for the details (including the definition of the Gaussian Unitary Ensemble) and the proofs.”
Ordered Random Walks (2006, background): “A survey on non-colliding random walks appears in [ K05 , Ch. 4].”
The Surprising Mathematics of Longest Increasing Subsequences (2015, background): “See the survey [72] for more information and many examples.”
Dyson's Brownian motions, intertwining and interlacing (2005, background): “A recent survey is given by König [13].”
Monte Carlo with determinantal point processes (2016, background): “[König, 2005] for an overview.”
A pedestrian's view on interacting particle systems, KPZ universality and random matrices (2008, background): “For further information we refer the reader to the surveys [1, 129, 60, 68, 125, 88] and the monograph [11].”