2D Ising model: correlation functions at criticality via Riemann-type boundary value problems

TL;DR

This paper reviews recent advances in understanding correlation functions at criticality in the 2D Ising model, using Riemann boundary value problems to derive explicit formulas and analyze scaling limits.

Contribution

It introduces a Riemann-type boundary value problem approach to derive explicit correlation formulas and analyze their convergence in the critical planar Ising model.

Findings

Explicit formulas for spin correlations in the infinite-volume limit.

Convergence of fermionic correlators, energy-density, and spin expectations as mesh size tends to zero.

Discussion of scaling limits and fusion rules for mixed correlators involving spins, disorders, and fermions.

Abstract

In this note we overview recent convergence results for correlations in the critical planar nearest-neighbor Ising model. We start with a short discussion of the combinatorics of the model and a definition of fermionic and spinor observables. After that, we illustrate our approach to spin correlations by a derivation of two classical explicit formulae in the infinite-volume limit. Then we describe the convergence results (as the mesh size tends to zero, in arbitrary planar domains) for fermionic correlators, energy-density and spin expectations. Finally, we discuss scaling limits of mixed correlators involving spins, disorders and fermions, and the classical fusion rules for them.