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

@article{Bleher2017ThePS, title={The Pfaffian Sign Theorem for the Dimer Model on a Triangular Lattice}, author={Pavel Bleher and Brad Elwood and Dra{\vz}en Petrovi{\'c}}, journal={Journal of Statistical Physics}, year={2017}, volume={171}, pages={400-426} }

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…

## 3 Citations

### Specific heat and partition function zeros for the dimer model on the checkerboard B lattice: Finite-size effects.

- PhysicsPhysical review. E
- 2019

The partition function of the dimer model on a 2M×2N checkerboard B lattice wrapped on a torus is analyzed and very unusual behavior of the partition function zeros and the specific heat of the Dimer model is found.

### The dimer and Ising models on Klein bottles

- Mathematics
- 2020

We study the dimer and Ising models on a planar weighted graph with periodic-antiperiodic boundary conditions, i.e. a graph $\Gamma$ in the Klein bottle $K$. Let $\Gamma_{mn}$ denote the graph…

### Dimer model: Full asymptotic expansion of the partition function

- Mathematics, Computer ScienceJournal of Mathematical Physics
- 2018

We give a complete rigorous proof of the full asymptotic expansion of the partition function of the dimer model on a square lattice on a torus for general weights $z_h, z_v$ of the dimer model and…

## References

SHOWING 1-10 OF 16 REFERENCES

### Dimer model on a triangular lattice.

- PhysicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2011

It is found that the dimer model on an M×N triangular lattice wrapped on a torus can be described by a conformal field theory having a central charge c=-2 and the shift exponent for the specific heat is found to depend on the parity of the number of lattice sites N along a given lattice axis.

### Classical dimers on the triangular lattice

- Physics
- 2002

We study the classical hard-core dimer model on the triangular lattice. Following Kasteleyn's fundamental theorem on planar graphs, this problem is soluble using Pfaffians. This model is particularly…

### The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice

- Mathematics
- 1961

### The Statistics of Dimers on a Lattice

- Mathematics
- 1961

The number of ways in which a finite quadratic lattice (with edges or with periodic boundary conditions) can be fully covered with given numbers of “horizontal” and “vertical” dimers is rigorously…

### Dimers on Surface Graphs and Spin Structures. II

- Mathematics
- 2007

In a previous paper [3], we showed how certain orientations of the edges of a graph Γ embedded in a closed oriented surface Σ can be understood as discrete spin structures on Σ. We then used this…

### Identities between dimer partition functions on different surfaces

- Mathematics
- 2016

It is shown that if G is a locally but not globally bipartite graph embedded in the Möbius strip, then Z(G˜) is equal to the square of Z( G).

### Dimer Statistics and Phase Transitions

- Physics
- 1963

After the introduction of the concept of lattice graph and a brief discussion of its role in the theory of the Ising model, a related combinatorial problem is discussed, namely that of the statistics…

### On the Theory of Pfaffian Orientations. I. Perfect Matchings and Permanents

- MathematicsElectron. J. Comb.
- 1999

It is proved that the generating function of the perfect matchings of a graph of genus $g$ may be written as a linear combination of Pfaffians, and a combinatorial way to compute the permanent of a square matrix is presented.

### Dimers on Surface Graphs and Spin Structures. I

- Mathematics
- 2007

Partition functions for dimers on closed oriented surfaces are known to be alternating sums of Pfaffians of Kasteleyn matrices. In this paper, we obtain the formula for the coefficients in terms of…

### Matchings in Graphs on Non-orientable Surfaces

- MathematicsJ. Comb. Theory, Ser. B
- 2000

“Crossing orientations,” the analogue of Kasteleyn's “admissible orientations” in this context, are introduced, describing how the Pfaffian of a signed adjacency matrix of a graph gives the sign of each perfect matching according to the number of edge-crossings in the matching.