Speaker: Kyle Petersen (Brandeis)
Title: Circles in Aztec Diamonds, Aztec Diamonds in Groves, Circles in Groves! Limiting behavior of two combinatorial models.
Abstract: An Aztec diamond of order n is a planar region defined as the union of all unit squares with integer vertices whose interiors lie inside the region |x+y| <= n+1. In earlier work, Jockusch, Propp, and Shor proved a theorem describing the limiting shape of a random domino tiling of an Aztec diamond. They showed the boundary between the uniformly tiled "frozen" corners and the more unpredictable "temperate zone" in the interior of the region approximated a circle. The so-called arctic circle theorem made precise the phenomenon observed in random tilings of large Aztec diamonds.
This talk will survey results for Aztec diamonds as well as examine a related combinatorial model called groves. Groves were created by Gabriel Carrol and David Speyer as a combinatorial interpretation for Laurent polynomials given by the cube recurrence. Groves have observable frozen regions which can be described precisely via asymptotic analysis of a generating function using methods due to Pemantle. These methods should also give a new proof of the arctic circle theorem for Aztec diamonds.