Planar Ising model at criticality: state-of-the-art and perspectives

TL;DR

This paper reviews recent progress on the conformal invariance of the critical planar Ising model, introduces s-embeddings for weighted graphs, and discusses potential pathways to universality results in this domain.

Contribution

It introduces s-embeddings as a new class of graph embeddings that could advance understanding of universality in the critical planar Ising model.

Findings

Review of conformal invariance developments since Smirnov's work

Introduction of s-embeddings for weighted planar graphs

Potential for establishing universality results in the Ising model

Abstract

In this essay, we briefly discuss recent developments, started a decade ago in the seminal work of Smirnov and continued by a number of authors, centered around the conformal invariance of the critical planar Ising model on $\mathbb{Z}^2$ and, more generally, of the critical Z-invariant Ising model on isoradial graphs (rhombic lattices). We also introduce a new class of embeddings of general weighted planar graphs (s-embeddings), which might, in particular, pave the way to true universality results for the planar Ising model.