Revisiting the combinatorics of the 2D Ising model

Abstract

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.

Circle Patterns and Critical Ising Models

“…which gives foundations for our results was proved in [13,27], and is valid for any weight vector x. We refer the interested reader to [11] for a recent detailed account of the relationship of (3.6) with other combinatorial approaches to the Ising model.…”

“…To be more precise, note that |F(c)| = |v − u| 1/2 is a well defined function on the corners in Υ that satisfies the three term relation (6.1) with coefficients multiplied byη(c) := F(c)/|F(c)|. This means that |F| is in the kernel of the (infinite) matrix D defined before Lemma 3.4 in [11]. We easily obtain from this lemma that 2D = C[I − 1 2 (Y + iI)D], where I is the identity, Y is the involutive automorphism induced by e → − e, and where C is an explicit conjugate of the (infinite) Kac-Ward matrix defined after equation (3.1) in [11].…”

Emergent planarity in two-dimensional Ising models with finite-range Interactions

“…In [40] these were linked with non-linear difference equations obeyed by the correlation functions of various combinations of order and disorder variables. Somewhat more specific statements on Pfaffian correlation functions from a combinatorial perspective can be found in the recent papers [8,11,36]. Our goal in this section is to present an elementary derivation of the Pfaffian structure of the correlations of paired order-disorder operators from the random current perspective.…”

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

“…Preliminaries. In this subsection, we recall some results of Loebl-Masbaum [14], Tesler [18] and Chelkak-Cimasoni-Kassel [1] as a preparation for the proofs of Theorems 2.2 and 2.3; the easy proofs are included for completeness. Let us first come back to the drawing of G and see how the intersection form (·) on H 1 (Σ; Z 2 ) relates to the one in the plane.…”

“…The latter quantity then can be proved combinatorially to be equal to the Ising high-temperature expansion twisted by signs, and therefore we end up with the general Kac-Ward formula. We also would like to recall that, by [1,Section 2.2], the signs of dimer configurations of G T in the expansion of Pf ( K λ ) amounts to saying that the spin structure λ is equivalent to a crossing orientation of G T as defined by Tesler [18]. Therefore it is possible to replace the generalised Kac-Ward matrix KW λ (G) by an adjacency matrix of G T with respect to the corresponding crossing orientation so that the general Kac-Ward formula for G boils down to a Pfaffian formula for G T .…”