The University of Michigan Combinatorics Seminar
We consider triangular holes on the hexagonal lattice and we
study their interaction when the rest of the lattice is covered by dimers.
More precisely, we analyze the joint correlation of these triangular holes
in a ``sea'' of dimers. We determine the asymptotics of the joint
correlation (for large separations between the holes) in the case when one
of the holes has side 1, all remaining holes have side 2, and the holes
are distributed symmetrically with respect to a symmetry axis. Our result
has a striking physical interpretation. If we regard the holes as
electrical charges, with charge equal to the difference between the number
of down-pointing and up-pointing unit triangles in a hole, the logarithm
of the joint correlation behaves exactly like the electrostatic potential
energy of this two-dimensional electrostatic system: it is obtained by a
Superposition Principle from the interaction of all pairs, and the pair
interactions are according to Coulomb's law.