Exact finite-size corrections in the dimer model on a cylinder

TL;DR

This paper derives exact finite-size corrections for the free energy of the dimer model on cylindrical lattices and analyzes the behavior of expansion coefficient ratios at various aspect ratios, revealing model-dependent critical phenomena.

Contribution

It provides the first exact finite-size correction formulas for the dimer model on cylinders with different boundary conditions and analyzes their asymptotic behavior.

Findings

Finite-size correction formulas derived for three lattice configurations.

Ratios of expansion coefficients exhibit abrupt changes at certain aspect ratios.

Critical aspect ratios and ratios vary between different models.

Abstract

The exact finite-size corrections to the free energy $F$ of the dimer model on lattice $\mathcal{M} \times \mathcal{N}$ with cylindrical boundary conditions have been derived for three cases where the lattice is completely covered by dimers: $\mathcal{M} = 2M$, $\mathcal{N} = 2N$; $\mathcal{M} = 2M - 1$, $\mathcal{N} = 2N$; and $\mathcal{M} = 2M$, $\mathcal{N} = 2N - 1$. For these types of cylinders, ratios $r_p(\rho)$ of the $p$th coefficient of $F$ have been calculated for the infinitely long cylinder (${\mathcal M} \rightarrow \infty$) and infinitely long strip (${\mathcal N} \rightarrow \infty$) at varying aspect ratios. As in previous studies of the dimer model on the rectangular lattice with free boundary conditions and for the Ising model with Brascamp-Kunz boundary conditions, the limiting values $p \to \infty$ exhibit abrupt anomalous behaviour of ratios $r_p(\rho)$ at certain values of $\rho$. These critical values of $\rho$ and the limiting values of the finite-size expansion coefficient ratios vary between the different models.