Journal of Statistical Physics

, Volume 171, Issue 3, pp 400–426 | Cite as

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

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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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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Indiana University–Purdue University IndianapolisIndianapolisUSA

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