Journal of Statistical Physics

, Volume 171, Issue 3, pp 400–426

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

• Pavel Bleher
• Dražen Petrović
Article

## 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.

## Keywords

Dimer model Exact solution Triangular lattice Periodic boundary conditions Pfaffian

## Notes

### Acknowledgements

The authors thank Barry McCoy and Dan Ramras for useful discussions, and the referee for a simplified proof of Theorem 5.2.

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