Revisiting the combinatorics of the 2D Ising model

TL;DR

This paper revisits the combinatorial formulas for the 2D Ising model's partition function and correlations, providing original proofs and exploring connections between different formalisms, with extensions to the double-Ising model.

Contribution

It offers original proofs of combinatorial formulas for the 2D Ising model and discusses their relations to various formalisms, extending results to the double-Ising model.

Findings

Provides original combinatorial formulas for the Ising model

Establishes connections between different formalisms

Extends formulas to the double-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.