Exact asymptotics of monomer-dimer model on rectangular semi-infinite lattices

TL;DR

This paper derives exact asymptotic expansions for the free energy of the monomer-dimer model on rectangular lattices, revealing dependence on lattice parity and matching infinite lattice behavior in low-density limits.

Contribution

It applies Pemantle and Wilson's asymptotic theory to obtain precise asymptotics for finite-width lattices, highlighting parity effects and series expansion similarities with infinite lattices.

Findings

Confirmed parity dependence of free energy at high dimer density

Derived identical first n terms in series expansion for low dimer density on finite and infinite lattices

Provided exact asymptotic formulas for small n in rectangular lattices

Abstract

By using the asymptotic theory of Pemantle and Wilson, exact asymptotic expansions of the free energy of the monomer-dimer model on rectangular $n \times \infty$ lattices in terms of dimer density are obtained for small values of $n$, at both high and low dimer density limits. In the high dimer density limit, the theoretical results confirm the dependence of the free energy on the parity of $n$, a result obtained previously by computational methods. In the low dimer density limit, the free energy on a cylinder $n \times \infty$ lattice strip has exactly the same first $n$ terms in the series expansion as that of infinite $\infty \times \infty$ lattice.