Elliptically distributed lozenge tilings of a hexagon

Details Elliptically distributed lozenge tilings of a hexagon Elliptically distributed lozenge tilings of a hexagon

2011-10-19

arxiv.org/abs/1110.4176v1

Physics

Mathematical Physics

47 pages, 19 figures

Scientific paper

We present a detailed study of a 4 parameter family of elliptic weights on tilings of a hexagon introduced by Borodin, Gorin and Rains, and generalize some of their results. In the process, we connect the combinatorics of the model with the theory of elliptic special functions. We first analyze some properties of the measure and introduce canonical coordinates that are useful for combinatorially interpreting results. We then show how the computed $n$-point function (called the elliptic Selberg density) and transitional probabilities connect to the theory of $BC_n$-symmetric multivariate elliptic special functions and difference operators discovered by Rains. In particular, the difference operators intrinsically capture the combinatorial model under study, while the elliptic Selberg density is a generalization (deformation) of probability distributions pervasive in the theory of random matrices and interacting particle systems. Based on quasi-commutation relations between elliptic difference operators, we construct certain natural measure-preserving Markov chains on such tilings. We then immediately obtain and describe an exact sampling algorithm from such distributions. We present sample random tilings from these measures showing an arctic boundary phenomenon. Interesting examples include a 1 parameter family of tilings where the arctic curve acquires 3 nodes. Finally, we show that the particle process associated to such tilings is determinantal with correlation kernel given in terms of the univariate elliptic biorthogonal functions of Spiridonov and Zhedanov.

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