Anthony Metcalfe: Universality classes of lozenge tilings of a polyhedron

Abstract:  A regular hexagon can be tiled with lozenges of three
different orientations. Letting the hexagon have sides of length n,
and the lozenges have sides of length 1, we can consider the
asymptotic behaviour of a typical tiling as n increases. Typically,
near the corners of the hexagon there are regions of "frozen" tiles,
and there is a "disordered" region in the centre which is
approximately circular.

More generally one can consider lozenge tilings of polyhedra with
more complex boundary conditions. In this talk we use
steepest descent analysis to examine the local asymptotic behaviour
of tiles in various regions. Tiles near the boundary of the
equivalent "frozen" and "disordered" regions are of particular
interest, and we give necessary conditions under which such tiles
behave asymptotically like a determinantal random point field
with the Airy kernel. We also classify necessary conditions that
lead to other asymptotic behaviours, and examine the global
asymptotic behaviour of the system by considering the geometric
implications of these conditions. 

This is joint work with Kurt Johansson and Erik Duse.