The University of Michigan Combinatorics Seminar
Fall 2004
October 15, 4:10-5:00, 3866 East Hall



Holes in random tilings with periodic boundary conditions
interact like charges in electrostatics

Mihai Ciucu

Georgia Tech


Abstract

In a talk given in this seminar about one year ago we defined the correlation of holes on the triangular lattice by including them in large hexagons that were grown to infinity so that the holes remained near the center. We showed that if the holes are distributed symetrically about a straight line, then for large distances between the holes the correlation behaves like the electrostatic energy of a two dimensional system of charges corresponding to the holes. Since the dimer statistics is significantly distorted almost everywhere inside hexagonal regions, it arises as a desirable goal to define the correlation of holes in an alternate way, via regions that don't distort dimer statistics, and analyze its asymptotic behavior. In the present talk we define such a correlation and prove that it also reduces to electrostatics in the scaling limit. Our proof applies to general, not necessarily symmetric distributions of the holes.