Non-intersecting paths and Hahn orthogonal polynomial ensemble

We compute the bulk limit of the correlation functions for the uniform measure on lozenge tilings of a hexagon. The limiting determinantal process is a translation invariant extension of the discrete sine process, which also describes the ergodic Gibbs measure of an appropriate slope. Introduction In this paper we study a well-known combinatorial model of determinantal point processes in dimension 1+1 (one spatial and one time variable). The model depends on 3 positive integers a, b, c and is given by the uniform distribution on the finite set Ω(a, b, c) of combinatorial objects that can be described in several equivalent ways: 3d Young diagrams (in other words plane partitions) in an a × b × c box; tilings of the hexagon of side lengths a, b, c, a, b, c by rhombi of three types; and a dimer model on the honeycomb lattice (see e.g. Cohn–LarsenPropp [CLP], Johansson [J], Johansson–Nordenstam [JN]). Correspondences between different models can be found in the appendix. Following Johansson, we use yet another model where Ω(a, b, c) is identified with the ensemble of non-intersecting polygonal paths on the plane lattice. The latter model leads to determinantal point processes (put it otherwise, random point configurations) varying over time. The determinantal property, which is of crucial importance, means that dynamical (i.e. space-time) correlation functions of the model can be obtained as minors of some matrix which is called dynamical correlation kernel. The aim of the present paper is to compute the asymptotics of this kernel in the “bulk limit” regime. For a fixed time moment our model provides a random point configuration on the one-dimensional lattice — the so-called Hahn orthogonal polynomial ensemble. The dynamical model can be described as a chain of such ensembles with varying parameters, thus it can be called a dynamical Hahn ensemble. The dynamical correlation kernel is the so-called extended Hahn kernel; it has been previously obtained by Johansson [J] and Johansson–Nordenstam [JN]. The Eynard-Metha theorem [EM] provides the basis for computations of the kernel. The main result of our paper is computation of the asymptotics of the extended Hahn kernel in the bulk limit regime. The answer is given by the