Monomer Pair Correlations

Abstract

In this paper we evaluate the monomer pair correlation along the diagonals (p, p + 1) and (p + 1, p) of a square lattice otherwise packed with dimers. Using the perturbed Pfaffian technique the correlation can be expressed as a Toeplitz determinant i^p |b_i−j+1|/2x generated by the function $$\mathrm{B(\theta )=}{\displaystyle \sum _{\mathrm{-\infty }}^{\infty }}{b}_k{e}^{\mathrm{ik\theta }}{\mathrm{=i\tau e}}^{\mathrm{i\theta }}\left[\frac{\mathrm{sgn}(\sin {\mathrm{\hspace{0.17em}\theta )+i\tau e}}^{\mathrm{i\theta }}}{{\text{1+iθe}}^{\mathrm{i\theta }}}\right]+{\displaystyle \sum _{\text{k=0}}^{\infty }}{\left(\frac{i}{\tau }\right)}^k{e}^{\mathrm{-ik\theta }}$$ , where τ = x/y and x and y are the activities of x and y dimers. We will calculate the determinant exactly and prove that the correlation decays with increasing monomer separation as B/4r^½; where B is simply related to the decay constant of the diagonal spin correlation at the critical point of a square ferromagnetic Ising lattice. The exact value is found to be B = 0.989487291.