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<p>We state and give two proofs for a Kac-Ward formula to compute the partition function of the Ising model on graphs embedded in closed, possibly non-orientable surfaces. The formula, which is based on the work of Cimasoni on dimers in non-orientable surfaces [<CitationRef CitationID="CR2">2</CitationRef>], is an extension of the Kac-Ward formula for planar graphs proved by Dolbilin et al [<CitationRef CitationID="CR5">5</CitationRef>] and the generalized Kac-Ward formula for graphs embedded in orientable surfaces obtained by Cimasoni [<CitationRef CitationID="CR3">3</CitationRef>]. It can be also considered as a different version of the Pfaffian formula given by the author in a recent work [<CitationRef CitationID="CR16">16</CitationRef>].</p>

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A Kac-Ward formula for the Ising partition function of surface graphs

  • 期刊论文

  • 发表日期：

    2025年12月9日

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  • Anh Minh Pham

摘要

We state and give two proofs for a Kac-Ward formula to compute the partition function of the Ising model on graphs embedded in closed, possibly non-orientable surfaces. The formula, which is based on the work of Cimasoni on dimers in non-orientable surfaces [2], is an extension of the Kac-Ward formula for planar graphs proved by Dolbilin et al [5] and the generalized Kac-Ward formula for graphs embedded in orientable surfaces obtained by Cimasoni [3]. It can be also considered as a different version of the Pfaffian formula given by the author in a recent work [16].