### Tuesday (Brandeis Monday) October 7, 2003

**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.