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Seminar

Elliptic distributions on stepped surfaces and an elliptic biorthogonal ensemble November 10, 2010
Parent Program: --
Location: MSRI: Baker Board Room
Description Speaker: Dan Betea

Abstract:
We will discuss a model of random tilings of a fixed hexagon (equivalently, boxed plane partitions or non-intersecting lattice paths of a special kind) where the probability distribution is a certain elliptic generalization of the uniform measure. Limits of this distribution have been studied by (among others) Borodin, Gorin, Rains, Kenyon and Okounkov. We will describe the combinatorics and algebra of this model and compute the n-point probability density of particles. This density is a top hypergeometric level elliptic generalization of discrete (Hahn, q-Racah), but also continuous (Hermite) ensembles. Based on this, we will construct a polynomial time Markov Chain algorithm for sampling from this distribution based on previous work of Diaconis and Fill/Borodin and Ferrari. At the core of the algorithm are certain quasi-commutation relations between elliptic difference operators discovered by Rains. Sampling from this algorithm, one can empirically observe an arctic boundary (frozen limit) behavior in the appropriate scaling limit. Viewed as a Markov process, such a random surface is a determinantal point process with correlation (Christoffel-Darboux) kernel given in terms of elliptic biorthogonal functions discovered by Spiridonov and Zhedanov. We may briefly survey methods of obtaining the frozen limit by analyzing asymptotics of such biorthogonal functions.



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