Asymptotics of block Toeplitz determinants and the classical dimer model

TL;DR

This paper analyzes the asymptotic behavior of block Toeplitz determinants related to the classical dimer model on a triangular lattice, providing explicit asymptotics for a range of parameters using advanced mathematical techniques.

Contribution

It extends the understanding of dimer models by computing explicit asymptotics of block Toeplitz determinants across a parameter range, including the challenging constant term evaluation.

Findings

Derived asymptotics for the monomer-monomer correlation function

Applied Szeg"o Limit Theorem to the block Toeplitz determinants

Successfully evaluated the constant term in the asymptotics

Abstract

We compute the asymptotics of a block Toeplitz determinant which arises in the classical dimer model for the triangular lattice when considering the monomer-monomer correlation function. The model depends on a parameter interpolating between the square lattice ($t=0$) and the triangular lattice ($t=1$), and we obtain the asymptotics for $0<t\le 1$. For $0<t<1$ we apply the Szeg\"o Limit Theorem for block Toeplitz determinants. The main difficulty is to evaluate the constant term in the asymptotics, which is generally given only in a rather abstract form.