The Pfaffian Sign Theorem for the Dimer Model on a Triangular Lattice

TL;DR

This paper proves the Pfaffian Sign Theorem for the dimer model on a triangular lattice, establishing the signs of Pfaffians for different boundary conditions and deriving asymptotics for the partition function.

Contribution

It establishes the Pfaffian Sign Theorem for the triangular lattice dimer model and provides asymptotic formulas for the partition function.

Findings

Pfaffian of the Kasteleyn matrix with periodic-periodic boundary conditions is negative.

Pfaffians with mixed boundary conditions are positive.

Derived asymptotics of the dimer partition function with small error.

Abstract

We prove the Pfaffian Sign Theorem for the dimer model on a triangular lattice embedded in the torus. More specifically, we prove that the Pfaffian of the Kasteleyn periodic-periodic matrix is negative, while the Pfaffians of the Kasteleyn periodic-antiperiodic, antiperiodic-periodic, and antiperiodic-antiperiodic matrices are all positive. The proof is based on the Kasteleyn identities and on small weight expansions. As an application, we obtain an asymptotics of the dimer model partition function with an exponentially small error term.