Nonintersecting paths and the Hahn orthogonal polynomial ensemble
The paper deals with a combinatorial model of determinantal point processes in dimension \(1+1\) (one spatial and one time variable). The model depends on three positive integer parameters \(a, b,\) and \(c\) and is given by the uniform distribution on a finite set \(\Omega(a, b, c)\) of combinatorial objects. The author uses another model, where \(\Omega(a, b, c)\) is identified with an ensemble of nonintersecting polygonal paths on the plane lattice. The latter model leads to determinantal point processes (putting it otherwise, random point configurations) varying over time. The determinantal property, which is of crucial importance, means that dynamic (i.e. space-time) correlation functions of the model can be obtainet as minors of some matrix, which is called the \textit{dynamic correlation kernel}. The aim of the present paper is to compute the asymptotics of this kernel in the various limit regimes. The main result of this paper is the computation of the asymptotics of the extended Hahn kernel in the bulk limit regime.
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