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

Appendices

Appendix A: Proof of Lemma 2.1

We have that

$$\begin{aligned} \sigma \cup \sigma '=\bigsqcup _{i=1}^{r}\gamma _i. \end{aligned}$$

(A.1)

Since each vertex in $\gamma _i$ is occupied by a dimer either from $\sigma$ or $\sigma ',$ and the dimers in $\sigma \cup \sigma '$ alternate, we conclude that each $\gamma _i$ is of even length.

By (2.15),

$$\begin{aligned} {\mathrm{Pf}\,}A=\sum _{\sigma \in \Sigma _{m,n}}{\mathrm{sgn}\,}(\sigma )w(\sigma ). \end{aligned}$$

(A.2)

If we enumerate the vertices on $\Gamma _{m,n}$, i.e. permute the set of vertices $V_{m,n},$ then by the well-known fact (see e.g. [5]) that for an arbitrary matrix P of order $mn\times mn,$

$$\begin{aligned} {\mathrm{Pf}\,}(PAP^{T})=\det (P){\mathrm{Pf}\,}(A), \end{aligned}$$

(A.3)

we get

$$\begin{aligned} {\mathrm{Pf}\,}\rho (A)=(-1)^{\rho }{\mathrm{Pf}\,}(A), \end{aligned}$$

(A.4)

where $\rho$ is some permutation on $V_{m,n}.$ Here $\rho (A)$ denotes a matrix A whose rows and columns have been permuted by $\rho .$ In other words,

$$\begin{aligned} {\mathrm{Pf}\,}A=\sum _{\sigma \in \Sigma _{m,n}}{\mathrm{sgn}\,}(\sigma )w(\sigma ) =\sum _{\sigma \in \Sigma _{m,n}}(-1)^{\rho }[{\mathrm{sgn}\,}(\sigma )]_{\rho }w(\sigma ), \end{aligned}$$

(A.5)

where $[{\mathrm{sgn}\,}(\sigma )]_{\rho }$ indicates the sign of $\sigma$ with respect to some new enumeration $\rho$ of vertices. From (A.5), we have that

$$\begin{aligned} {\mathrm{sgn}\,}(\sigma )=(-1)^{\rho }[{\mathrm{sgn}\,}(\sigma )]_{\rho }. \end{aligned}$$

(A.6)

If we take any two configurations $\sigma$ and $\sigma '$, then (A.6) implies that

$$\begin{aligned} {\mathrm{sgn}\,}(\sigma )\cdot {\mathrm{sgn}\,}(\sigma ')=[{\mathrm{sgn}\,}(\sigma )]_{\rho }\cdot [{\mathrm{sgn}\,}(\sigma ')]_{\rho }, \end{aligned}$$

(A.7)

i.e. the sign of $\sigma \cup \sigma '$ is invariant under any renumeration of vertices.

Let

$$\begin{aligned} \rho = \left( \begin{matrix} 1 &{} 2 &{} \cdots &{} n_1 &{} n_1+1 &{} \cdots &{} n_1+n_2 &{} \cdots &{} n_1+\ldots +n_{r-1}+n_r \\ v_{1,1} &{} v_{1,2} &{} \cdots &{} v_{1,n_1} &{} v_{2,1} &{} \cdots &{} v_{2,n_2} &{} \cdots &{} v_{r,n_r} \end{matrix}\right) , \end{aligned}$$

(A.8)

where $v_{j,k}\in \{1,2,\ldots ,mn\}$ denotes the k-th vertex of the j-th contour. Note that $\rho$ is a renumeration of vertices so that along each $\gamma _i$ they are rearranged in a cyclical order, starting from one contour and continuing to the next one. Now, the underlying permutations $\pi (\sigma )$ and $\pi (\sigma ')$ with respect to $\rho$ are then:

$$\begin{aligned}&[\pi (\sigma )]_{\rho }=\mathrm{Id},\end{aligned}$$

(A.9)

$$\begin{aligned}&[\pi (\sigma ')]_{\rho }=\prod _{i=1}^{r}C(\gamma _i), \end{aligned}$$

(A.10)

where

$$\begin{aligned} C(\gamma _i) = \left( \begin{matrix} v_{i,1} &{} \cdots &{} v_{i,n_i} \\ v_{i,2} &{} \cdots &{} v_{i,1} \end{matrix}\right) . \end{aligned}$$

(A.11)

From this equation, Eq. (A.7), and the fact that each $\gamma _i$ corresponds to a cycle of even length, Lemma 2.1 follows.

Appendix B: Numerical Data for the Pfaffians $A_i$

In this Appendix we present numerical data for the Pfaffians $A_i$ on the $m\times n$ lattices on the torus for different values m and n. It is interesting to compare these data with the asymptotics of the Pfaffians $A_i$, obtained in Sects. 6 and 7 above, and also with the identities ${\mathrm{Pf}\,}A_1=-{\mathrm{Pf}\,}A_2$, ${\mathrm{Pf}\,}A_3={\mathrm{Pf}\,}A_4$ for odd n, proven in Sect. 5.

The Pfaffians $A_i$ for $m=4$, $n=3$.

$$\begin{aligned} \begin{aligned} {\mathrm{Pf}\,}A_1&=-4z_hz_d(3z_v^2+z_d^2)(4z_h^2+3z_v^2+3z_d^2),\\ {\mathrm{Pf}\,}A_2&=4z_hz_d(3z_v^2+z_d^2)(4z_h^2+3z_v^2+3z_d^2),\\ {\mathrm{Pf}\,}A_3&=2(z_h^2+z_d^2)[(2z_h^2+3z_v^2)(2z_h^2+3z_v^2+4z_d^2)+z_d^4],\\ {\mathrm{Pf}\,}A_4&=2(z_h^2+z_d^2)[(2z_h^2+3z_v^2)(2z_h^2+3z_v^2+4z_d^2)+z_d^4]. \end{aligned} \end{aligned}$$

The Pfaffians $A_i$ for $m=4$, $n=4$.

$$\begin{aligned} \begin{aligned} {\mathrm{Pf}\,}A_1&=-256z_h^2z_v^2z_d^2(z_h^2+z_v^2+z_d^2),\\ {\mathrm{Pf}\,}A_2&=16(z_v^2+z_d^2)^2(2z_h^2+z_v^2+z_d^2)^2,\\ {\mathrm{Pf}\,}A_3&=16(z_v^2+z_d^2)^2(z_h^2+2z_v^2+z_d^2)^2,\\ {\mathrm{Pf}\,}A_3&=16(z_v^2+z_d^2)^2(z_h^2+z_v^2+2z_d^2)^2. \end{aligned} \end{aligned}$$

The Pfaffians $A_i$ for $m=4$, $n=6$.

$$\begin{aligned} \begin{aligned} {\mathrm{Pf}\,}A_1&=-16z_h^2z_d^2(4z_h^2+3z_v^2+3z_d^2)^2(3z_v^2+z_d^2)^2,\\ {\mathrm{Pf}\,}A_2&=16z_v^2(4z_h^2+z_v^2+z_d^2)^2(z_h^2+z_v^2+z_d^2)(z_v^2+3z_d^2)^2,\\ {\mathrm{Pf}\,}A_3&=4(z_d^2 + z_h^2)^2(z_d^4 + 4z_d^2 (2z_h^2 + 3z_v^2) + (2z_h^2 + 3z_v^2)^2)^2,\\ {\mathrm{Pf}\,}A_4&=4(z_d^2 + z_h^2 + 2z_v^2)^2(z_d^4 + 8z_d^2z_h^2 + 4z_h^4 + 4z_d^2z_v^2 + 4z_h^2z_v^2 + z_v^4)^2. \end{aligned} \end{aligned}$$

The Pfaffians $A_i$ for $m=4$, $n=8$.

$$\begin{aligned} \begin{aligned} {\mathrm{Pf}\,}A_1&=-4096z_d^2z_h^2z_v^2(z_d^2 + z_v^2)^2(z_d^2 + z_h^2 + z_v^2)(z_d^2 + 2z_h^2 + z_v^2)^2,\\ {\mathrm{Pf}\,}A_2&=16(z_d^4 + 6z_d^2z_v^2 + z_v^4)^2(z_d^4 + 8z_h^4 + 8z_h^2 z_v^2 + z_v^4 + 2z_d^2 (4z_h^2 + z_v^2))^2,\\ {\mathrm{Pf}\,}A_3&=256(z_d^2 + z_h^2)^2(z_h^2 + z_v^2)^2(2z_d^2 + z_h^2 + z_v^2)^2(z_d^2 + z_h^2 +2z_v^2)^2,\\ {\mathrm{Pf}\,}A_4&=16(z_d^4 + 2z_h^4 + 4z_h^2z_v^2 + z_v^4 + 2z_d^2(2z_h^2 + z_v^2))^2(z_d^4 + 2z_h^4 + 4z_h^2z_v^2\\&\quad + z_v^4 + z_d^2 (4z_h^2 + 6z_v^2))^2. \end{aligned} \end{aligned}$$

The Pfaffians $A_i$ for $m=6$, $n=6$.

$$\begin{aligned} \begin{aligned} {\mathrm{Pf}\,}A_1&=-4z_d^2(z_d^2 + 3z_h^2)^2(z_d^2 + 3z_v^2)^2 (z_d^2 + 3 (z_h^2 + z_v^2))^2 (4z_d^2 + 3 (z_h^2 + z_v^2))^2,\\ {\mathrm{Pf}\,}A_2&=4 z_v^2 (3 z_d^2 + z_v^2)^2 (3 z_h^2 + z_v^2)^2 (3 (z_d^2 + z_h^2) + z_v^2)^2 (3 (z_d^2 + z_h^2) + 4 z_v^2)^2,\\ {\mathrm{Pf}\,}A_3&=4 z_h^2 (3 z_d^2 + z_h^2)^2 (z_h^2 + 3 z_v^2)^2 (3 z_d^2 + z_h^2 + 3 z_v^2)^2 (3 z_d^2 + 4 z_h^2 + 3 z_v^2)^2,\\ {\mathrm{Pf}\,}A_4&=4 (z_d^2 + z_h^2 + z_v^2)^3 (4 z_d^2 + z_h^2 + z_v^2)^2 (z_d^2 + 4 z_h^2 + z_v^2)^2 (z_d^2 + z_h^2 + 4 z_v^2)^2. \end{aligned} \end{aligned}$$

The Pfaffians $A_i$ for $m=8$, $n=8$.

$$\begin{aligned} \begin{aligned} {\mathrm{Pf}\,}A_1&=-1048576 z_h^2z_v^2z_d^2 (z_d^2 + z_h^2)^2 (z_d^2 + z_v^2)^2 (z_h^2 + z_v^2)^2 (z_d^2 + z_h^2 + z_v^2) (2 z_d^2 + z_h^2 + z_v^2)^2 \\&\quad \times (z_d^2 + 2 z_h^2 + z_v^2)^2 (z_d^2 +z_h^2 + 2z_v^2)^2,\\ {\mathrm{Pf}\,}A_2&=256(z_d^4 +6z_d^2z_v^2 + z_v^4)^2(z_d^4 + 4z_d^2z_h^2 + 2z_h^4 + 2z_d^2z_v^2 + 4z_h^2z_v^2 +z_v^4)^2\\&\quad \times (z_d^4 + 4z_d^2z_h^2 + 2z_h^4 + 6z_d^2z_v^2 + 4z_h^2z_v^2 + z_v^4)^2\\&\quad \times (z_d^4+ 8z_d^2z_h^2 + 8z_h^4 + 2z_d^2z_v^2 + 8z_h^2z_v^2 +z_v^4)^2,\\ {\mathrm{Pf}\,}A_3&=256(z_d^4 + 6z_d^2z_h^2 + z_h^4)^2(z_d^4 + 2z_d^2z_h^2 + z_h^4 + 4z_d^2z_v^2 + 4z_h^2 z_v^2 + 2z_v^4)^2\\&\quad \times (z_d^4 + 6z_d^2z_h^2 + z_h^4 + 4z_d^2z_v^2 + 4z_h^2z_v^2 + 2z_v^4)^2 (z_d^4 + 2z_d^2z_h^2 + z_h^4\\&\quad + 8z_d^2z_v^2 + 8z_h^2z_v^2 + 8z_v^4)^2,\\ {\mathrm{Pf}\,}A_4&=256 (2 z_d^4 + 4 z_d^2 z_h^2 + z_h^4 + 4 z_d^2 z_v^2 + 2 z_h^2 z_v^2 + z_v^4)^2\\&\quad \times (8 z_d^4+ 8 z_d^2 z_h^2 + z_h^4 + 8 z_d^2 z_v^2 + 2 z_h^2 z_v^2 +z_v^4)^2 \\&\quad \times (z_h^4 + 6 z_h^2 z_v^2 + z_v^4)^2 (2 z_d^4 + 4 z_d^2 z_h^2 + z_h^4 +4 z_d^2 z_v^2 + 6 z_h^2 z_v^2 + z_v^4)^2. \end{aligned} \end{aligned}$$