Eric Rains (CalTech)
Lozenge tilings and elliptic biorthogonal functions
One of the many combinatorial models in which fluctuations are controlled by random-matrix-related
distributions is that of uniform random lozenge tilings of hexagons (or, equivalently, uniform random
plane partitions in a box). These exhibit an "arctic circle" phenomenon--an inscribed ellipse outside of
which the tiling can be reliably predicted--and the fluctuations in said ellipse are controlled by the TracyWidom distribution. The analysis of the uniform case rests on the fact that the tiling model can be
viewed as a determinantal process with kernel expressed via Hahn polynomials. I'll discuss a recent
generalization of this (joint with Borodin and Gorin), in which a suitable weighting of the probabilities by
elliptic functions gives a process related to elliptic biorthogonal functions and degenerations thereof
(e.g., q-Racah polynomials).