Rotational invariance of quadromer correlations on the hexagonal lattice

In 1963 Fisher and Stephenson \cite{FS} conjectured that the monomer-monomer correlation on the square lattice is rotationally invariant. In this paper we prove a closely related statement on the hexagonal lattice. Namely, we consider correlations of two quadromers (four-vertex subgraphs consisting of a monomer and its three neighbors) and show that they are rotationally invariant.Comment: 28 page

Macroscopically separated gaps in dimer coverings of Aztec rectangles

In this paper we determine the interaction of diagonal defect clusters in regions of an Aztec rectangle that scale to arbitrary points on its symmetry axis (in earlier work we treated the case when this point was the center of the scaled Aztec rectangle). We use the resulting formulas to determine the asymptotics of the correlation of defects that are macroscopically separated from one another and feel the influence of the boundary. In several of the treated situations this seems not to be accomplishable by previous methods. Our applications include the case of two long neutral strings, which turn out to interact by an analog of the Casimir force, two families of neutral doublets that turn out to interact completely independently of one another, a neutral doublet and a very long neutral string, a general collection of macroscopically separated monomer and separation defects, and the case of long strings consisting of consecutive monomers.Comment: 43 page

The equivalence between enumerating cyclically symmetric, self-complementary and totally symmetric, self-complementary plane partitions

We prove that the number of cyclically symmetric, self-complementary plane partitions contained in a cube of side $2n$ equals the square of the number of totally symmetric, self-complementary plane partitions contained in the same cube, without explicitly evaluating either of these numbers. This appears to be the first direct proof of this fact. The problem of finding such a proof was suggested by Stanley