2020
DOI: 10.48550/arxiv.2010.11047
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The dimer and Ising models on Klein bottles

Abstract: We study the dimer and Ising models on a planar weighted graph with periodicantiperiodic boundary conditions, i.e. a graph Γ in the Klein bottle K . Let Γmn denote the graph obtained by pasting m rows and n columns of copies of Γ, which embeds in K for n odd and in the torus T 2 for n even. We compute the dimer partition function Zmn of Γmn for n odd, in terms of the well-known characteristic polynomial P of Γ12 ⊂ T 2 together with a new characteristic polynomial R of Γ ⊂ K .Using this result together with wor… Show more

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Cited by 2 publications
(4 citation statements)
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References 33 publications
(128 reference statements)
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“…It turns out that they split as products of representations of degree 1 and 2, yielding a closed formula for Z dimer (Γ mn , x) in terms of determinants of A τ Γ , with τ of degree 1 and 2. This result is at the core of the study of the dimer model on Klein bottles of the first-named author [9].…”
Section: 1mentioning
confidence: 66%
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“…It turns out that they split as products of representations of degree 1 and 2, yielding a closed formula for Z dimer (Γ mn , x) in terms of determinants of A τ Γ , with τ of degree 1 and 2. This result is at the core of the study of the dimer model on Klein bottles of the first-named author [9].…”
Section: 1mentioning
confidence: 66%
“…Applying Corollary 3.9 to the case of a graph embedded in the torus yields an immediate proof of the classical fact that the dimer characteristic polynomial behaves multiplicatively under so-called enlargement of the fundamental domain [24,Theorem 3.3]. However, applying our results to the study of the dimer model on graphs embedded in the Klein bottle leads to new powerful results, that are harnessed in the parallel article [9].…”
Section: Whenever a ρmentioning
confidence: 78%
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“…There are a number of interesting problems which could be studied if this limitation could be overcome. In the last few years, similar expansions for periodic graphs imbedded in the torus [KSW16] or Klein bottle [Cim20] based on the characteristic polynomial method have found a constant term which is universal (i.e. independent of many of the details of exactly which graph is chosen); it seems reasonable to expect that the same is true for some class of planar graphs.…”
Section: Introductionmentioning
confidence: 99%