Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model

TL;DR

This paper constructs discrete holomorphic observables in the critical Ising model, demonstrating their conformally covariant scaling limits and establishing the model as conformally invariant in a rigorous manner.

Contribution

It introduces the first rigorous proof of conformal invariance in the Ising model through discrete holomorphic observables and their scaling limits.

Findings

Discrete holomorphic observables are constructed at criticality.

Scaling limits of observables are conformally covariant.

First rigorous establishment of conformal invariance in the Ising model.

Abstract

We construct discrete holomorphic observables in the Ising model at criticality and show that they have conformally covariant scaling limits (as mesh of the lattice tends to zero). In the sequel those observables are used to construct conformally invariant scaling limits of interfaces. Though Ising model is often cited as a classical example of conformal invariance, it seems that ours is the first paper where it is actually established.