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Browsing by Author "Elwood, Brad"
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Item Dimer model: Full asymptotic expansion of the partition function(AIP, 2018) Bleher, Pavel; Elwood, Brad; Petrović, Dražen; Mathematical Sciences, School of ScienceWe 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 zh, zv of the dimer model and arbitrary dimensions of the lattice m, n. We assume m is even and we show that the asymptotic expansion depends on the parity of n. We review and extend the results of Ivashkevich et al. [J. Phys. A: Math. Gen. 35, 5543 (2002)] on the full asymptotic expansion of the partition function of the dimer model, and we give a rigorous estimate of the error term in the asymptotic expansion of the partition function.Item The Pfaffian Sign Theorem for the Dimer Model on a Triangular Lattice(Springer, 2018) Bleher, Pavel; Elwood, Brad; Petrović, Dražen; Mathematical Sciences, School of ScienceWe 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.