The two-dimensional Ising model and the Kac-Ward determinant

Долбилин, Николай Петрович; Zinov'ev, Yu. M.; Mishchenko, А. S.; Shtan'ko, M A; Штогрин, Михаил Иванович

doi:10.1070/im1999v063n04abeh000251

Cited by 18 publications

(21 citation statements)

References 4 publications

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“…1a. and it was an intricate story [98,104,28,60,42] to give a fully rigorous proof of this identity for general planar graphs; see a recent paper [76] by Lis for a streamlined version of the classical approach to (1.5). This formula was generalized to graphs embedded in surfaces in [22].…”

Section: The Kac-ward Matrix and The Terminal Graph Letmentioning

confidence: 99%

Revisiting the combinatorics of the 2D Ising model

Chelkak¹,

Cimasoni²,

Kassel³

2017

Ann. Inst. Henri Poincaré Comb. Phys. Interact.

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We provide a concise exposition with original proofs of combinatorial formulas for the 2D Ising model partition function, multi-point fermionic observables, spin and energy density correlations, for general graphs and interaction constants, using the language of Kac-Ward matrices. We also give a brief account of the relations between various alternative formalisms which have been used in the combinatorial study of the planar Ising model: dimers and Grassmann variables, spin and disorder operators, and, more recently, s-holomorphic observables. In addition, we point out that these formulas can be extended to the double-Ising model, defined as a pointwise product of two Ising spin configurations on the same discrete domain, coupled along the boundary.2000 Mathematics Subject Classification. 82B20.

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Section: The Kac-ward Matrix and The Terminal Graph Letmentioning

confidence: 99%

Revisiting the combinatorics of the 2D Ising model

Chelkak¹,

Cimasoni²,

Kassel³

2017

Ann. Inst. Henri Poincaré Comb. Phys. Interact.

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“…where C(ω) is the number of pairs of edges in ω that cross. The seminal identity of Kac and Ward [22] (which in the general form allowing edge crossings was established in [15,23,31]) reads…”

Section: The Kac-ward Operator and The Fermionic Observablementioning

confidence: 99%

Circle Patterns and Critical Ising Models

Lis

2019

Commun. Math. Phys.

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“…Hence together with an idea of Loebl-Masbaum [14], we get an elementary, self-contained proof for the Pfaffian formula. Let us also mention that using the method of Dolbilin et al [5] (and then developed by Cimasoni [3]) one can have a Kac-Ward formula for graphs embedded in (possibly) non-orientable surfaces. In this case generalised Kac-Ward matrices can be encoded using pin − structures, which generalise spin structures to non-orientable surfaces [13].…”

Section: Introductionmentioning

confidence: 99%

A Pfaffian formula for the monomer–dimer model on surface graphs

Phạm

2018

Lett Math Phys

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We give a Pfaffian formula to compute the partition function of the Ising model on any graph G embedded in a closed, possibly non-orientable surface. This formula, which is suitable for computational purposes, is based on the relation between the Ising model on G and the dimer model on its terminal graph G T . By combining the ideas of Loebl-Masbaum [14], Tesler [18], Cimasoni [2, 3] and Chelkak-Cimasoni-Kassel [1], we give an elementary proof for the formula.2010 Mathematics Subject Classification. Primary 82B20; Secondary 05C70, 05C10, 57M15.

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The two-dimensional Ising model and the Kac-Ward determinant

Cited by 18 publications

References 4 publications

Revisiting the combinatorics of the 2D Ising model

Revisiting the combinatorics of the 2D Ising model

Circle Patterns and Critical Ising Models

A Pfaffian formula for the monomer–dimer model on surface graphs

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