Conformal invariance of lattice models

TL;DR

This paper discusses the conformal invariance properties of planar critical lattice models, focusing on the Ising and FK-Ising models, and explores discrete holomorphic functions and their applications to statistical physics.

Contribution

It provides a comprehensive, self-contained account of conformal invariance in lattice models, emphasizing discrete holomorphic functions and convergence to SLE.

Findings

Convergence of fermionic observables established

Application of discrete holomorphic functions to statistical physics

Discussion of open questions in conformal invariance

Abstract

These lecture notes provide a (almost) self-contained account on conformal invariance of the planar critical Ising and FK-Ising models. They present the theory of discrete holomorphic functions and its applications to planar statistical physics (more precisely to the convergence of fermionic observables). Convergence to SLE is discussed briefly. Many open questions are included.