Universality in the 2D Ising model and conformal invariance of fermionic observables

TL;DR

This paper proves that the critical 2D Ising model exhibits universal and conformally invariant scaling limits by introducing discrete holomorphic fermions, confirming long-standing conjectures in mathematical physics.

Contribution

It provides the first rigorous proof of universality and conformal invariance for the critical 2D Ising model using discrete holomorphic fermions on various planar graphs.

Findings

Discrete holomorphic fermions have universal scaling limits.

The scaling limits are conformally invariant.

The results confirm the universality conjecture for the 2D Ising model.

Abstract

It is widely believed that the celebrated 2D Ising model at criticality has a universal and conformally invariant scaling limit, which is used in deriving many of its properties. However, no mathematical proof of universality and conformal invariance has ever been given, and even physics arguments support (a priori weaker) M\"obius invariance. We introduce discrete holomorphic fermions for the 2D Ising model at criticality on a large family of planar graphs. We show that on bounded domains with appropriate boundary conditions, those have universal and conformally invariant scaling limits, thus proving the universality and conformal invariance conjectures.