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

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 periodicantiperiodic, 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 asymptotic behavior of the dimer model partition function with an exponentially small error term.

Dimer model: Full asymptotic expansion of the partition function

“…Lemma 3.1. As n, m → ∞ under condition (2.1), we have that G (2) m,n admits the following asymptotic expansion:…”

“…Using the Euler-Maclaurin formula (C.5), we obtain that G (2) n is expanded in the asymptotic series in powers of 1 n as…”

“…Consider now the second term in (3.49). Using estimate (3.47), we obtain that j,k: j,k≤R ln n; (2k+1)j>ξ Next, we would like to replace R ln n in B (2) n,K in (3.40) by ∞. Denote…”

The dimer and Ising models on Klein bottles

“…vertical) edges-weights equal to x (resp. y), periodicperiodic boundary conditions and mn even, we have log Z mn = mn f 0 (x, y) + fsc (−1) m+n ( nx my ) + o (1) , where the bulk free energy f 0 depends on the weights x, y in an explicit but complicated way, while the constant order finite size correction term fsc ± only depends on the shape parameter nx my of the torus, together with the parity of m + n. For conformally invariant two-dimensional models on a closed surface Σ of vanishing Euler characteristic, such an asymptotic expansion is believed to hold for arbitrary graphs, with the finite-size correction term depending only on the universality class of the model at criticality and on the topology of the surface, but not on the underlying graph [2,4]. There are exactly two closed surfaces with χ(Σ) = 0, namely the torus T 2 and the Klein bottle K , corresponding to periodic-periodic and periodic-antiperiodic boundary conditions, respectively.…”

“…(Here, we make use of the notation of [31], where P stands for the full polynomial which factors as P (z, w) = Q(z, w)Q(z −1 , w −1 ).) In particular, the Pfaffian formula of [45] can now be reformulated as (1) Z = 1 2 ±P (1, 1) 1/2 ± P (−1, 1) 1/2 ± P (1, −1) 1/2 ± P (−1, −1) 1/2 , where the signs can be given a natural geometric interpretation [13]. A crucial role is played by the intersection of the corresponding spectral curve, i.e.…”