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.

Minor change in the notation (definition of eta). 55 pages, 4 figures

References in corpus (9)

• Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model
• Smirnov's observable for free boundary conditions, interfaces and crossing probabilities
• A note on Ising random currents, Ising-FK, loop-soups and the Gaussian free field
• The scaling limit of the energy correlations in non integrable Ising models
• Duality between Spin networks and the 2D Ising model
• Scaling limits for the critical Fortuin-Kasteleyn model on a random planar map I: cone times
• XOR-Ising Loops and the Gaussian Free Field
• Level Lines of Gaussian Free Field I: Zero-Boundary GFF
• Bipartite dimer representation of squared 2d-Ising correlations

Cited by in corpus (14)

• Emergent Planarity in two-dimensional Ising Models with finite-range Interactions
• The planar Ising model and total positivity
• Discrete stress-energy tensor in the loop O(n) model
• Energy correlations of non-integrable Ising models: The scaling limit in the cylinder
• Non-integrable Ising models in cylindrical geometry: Grassmann representation and infinite volume limit
• Correlations of primary fields in the critical Ising model
• Derivation of the free energy, entropy and specific heat for planar Ising models: Application to Archimedean lattices and their duals
• Kac-Ward formula and its extension to order-disorder correlators through a graph zeta function
• A Pfaffian formula for the Ising partition function of surface graphs
• Slit-strip Ising boundary conformal field theory 2: Scaling limits of fusion coefficients
• The scaling limit of boundary spin correlations in non-integrable Ising models
• Bell polynomials in the series expansions of the Ising model
• The boundary disorder correlation for the Ising model on a cylinder
• Zero-temperature stochastic Ising model on planar quasi-transitive graphs