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

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 ($\mathcal{M}\to \infty$) and infinitely long strip ($\mathcal{N}\to \infty$) 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.

Supplementary material included with the paper

References in corpus (4)

• Exact solution of the $2d$ dimer model: Corner free energy, correlation functions and combinatorics
• Non-Local Finite-Size Effects in the Dimer Model
• Finite-size corrections and scaling for the dimer model on the checkerboard lattice
• Exact coefficients of finite-size corrections in the Ising model with Brascamp-Kunz boundary conditions and their relationships for strip and cylindrical geometries