Asymptotics of block Toeplitz determinants and the classical dimer model

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 Szegö 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.

References in corpus (5)

• An RVB phase in the triangular lattice quantum dimer model
• Short-ranged RVB physics, quantum dimer models and Ising gauge theories
• Three dimensional resonating valence bond liquids and their excitations
• Classical dimers on the triangular lattice
• On Ising and dimer models in two and three dimensions

Cited by in corpus (8)

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• Quantum fluctuations of one-dimensional free fermions and Fisher-Hartwig formula for Toeplitz determinants
• Block Toeplitz determinants, constrained KP and Gelfand-Dickey hierarchies
• Topological sectors, dimer correlations and monomers from the transfer-matrix solution of the dimer model
• Exact solution of the classical dimer model on a triangular lattice: Monomer-monomer correlations
• Breakdown of the thermodynamic limit in quantum spin and dimer models
• Spin-spin correlators on the $\beta$/${\beta }^{\star }$ boundaries in 2D Ising-like models: exact analysis through theory of block Toeplitz determinants
• Exactly Solvable Topological Phase Transition in a Quantum Dimer Model