Methods and Applications of Analysis

Volume 20 (2013)

Number 3

An ensemble related to discrete orthogonal polynomials and its application to tilings of a half-hexagon

Pages: 273 – 294

DOI: http://dx.doi.org/10.4310/MAA.2013.v20.n3.a4

Author

Uwe Schwerdtfeger (Fakultät für Mathematik, Technische Universität Chemnitz, Germany)

Abstract

We study a family of discrete probability measures that may be used to model particle configurations on a set of discrete nodes in presence of an impenetrable wall. The correlation functions are shown to be determinantal and can be expressed in terms of discrete orthogonal polynomials. We make strong use of the results of the monograph [1] to explain the asymptotic behaviour of these ensembles. Our general results are applied to random tilings of the half-hexagon, a model introduced in [4]. In regions away from the bottom boundary the model exhibits the same asymptotic behaviour as tilings of the full hexagon, including an “arctic half-ellipse.” Close to that boundary, however, the statistics are different.

Keywords

interacting particle systems, interacting random processes, discrete orthogonal polynomials, random tiling, non-intersecting lattice paths

2010 Mathematics Subject Classification

60K35, 82C22

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