Conformal invariance of crossing probabilities for the Ising model with free boundary conditions

TL;DR

This paper proves that crossing probabilities in the critical planar Ising model with free boundary conditions are conformally invariant in the scaling limit, using exploration processes converging to a specific SLE variant.

Contribution

It establishes conformal invariance of crossing probabilities for the Ising model with free boundaries and constructs an exploration tree analogous to Sheffield's for these conditions.

Findings

Crossing probabilities are conformally invariant in the scaling limit.

Exploration processes converge to SLE(3,-3/2,-3/2).

An exploration tree for free boundary conditions is constructed.

Abstract

We prove that crossing probabilities for the critical planar Ising model with free boundary conditions are conformally invariant in the scaling limit, a phenomenon first investigated numerically by Langlands, Lewis and Saint-Aubin. We do so by establishing the convergence of certain exploration processes towards SLE$(3,\frac{-3}2,\frac{-3}2)$. We also construct an exploration tree for free boundary conditions, analogous to the one introduced by Sheffield.