Monomer correlations on the square lattice

Combinatorics Seminar
Friday, April 9, 2010 - 15:05
1 hour (actually 50 minutes)
Skiles 255
Professor, Indiana University, Bloomington
In 1963 Fisher and Stephenson conjectured that the correlation function of two oppositely colored monomers in a sea of dimers on the square lattice is rotationally invariant in the scaling limit. More precisely, the conjecture states that if one of the monomers is fixed and the other recedes to infinity along a fixed ray, the correlation function is asymptotically $C d^(-1/2)$, where $d$ is the Euclidean distance between the monomers and $C$ is a constant independent of the slope of the ray. Shortly afterward Hartwig rigorously determined $C$ when the ray is in a diagonal direction, and this remains the only direction settled in the literature. We generalize Hartwig's result to any finite collection of monomers along a diagonal direction. This can be regarded as a counterpart of a result of Zuber and Itzykson on n-spin correlations in the Ising model. A special case proves that two same-color monomers interact the way physicists predicted.