2002/12/10 by Jinho Baik, Thomas Kriecherbauer, Baik, Jinho +6
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Classical Analysis and ODEs (math.CA) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical functions and polynomials #Probability (math.PR) #Random Matrices and Applications #math.CA #math.PR #nlin.SI
paper · pdf · doi:10.48550/arxiv.math/0212149
28 pages, 7 figures
arxiv created 2002/12/10 · openalex publication_date 2002/12/10 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We compute the pointwise asymptotics of orthogonal polynomials with respect to a general class of pure point measures supported on finite sets as both the number of nodes of the measure and also the degree of the orthogonal polynomials become large. The class of orthogonal polynomials we consider includes as special cases the Krawtchouk and Hahn classical discrete orthogonal polynomials, but is far more general. In particular, we consider nodes that are not necessarily equally spaced. The asymptotic results are given with error bound for all points in the complex plane except for a finite union of discs of arbitrarily small but fixed radii. These exceptional discs are the neighborhoods of the so-called band edges of the associated equilibrium measure. As applications, we prove universality results for correlation functions of a general class of discrete orthogonal polynomial ensembles, and in particular we deduce asymptotic formulae with error bound for certain statistics relevant in the random tiling of a hexagon with rhombus-shaped tiles. The discrete orthogonal polynomials are characterized in terms of a a Riemann-Hilbert problem formulated for a meromorphic matrix with certain pole conditions. By extending the methods of [17, 22], we suggest a general and unifying approach to handle Riemann-Hilbert problems in the situation when poles of the unknown matrix are accumulating on some set in the asymptotic limit of interest.