Limit of the Wulff Crystal when approaching criticality for site percolation on the triangular lattic

Hugo Duminil-Copin

doi:10.1214/ecp.v18-3163

Abstract

The understanding of site percolation on the triangular lattice progressed greatly in the last decade. Smirnov proved conformal invariance of critical percolation, thus paving the way for the construction of its scaling limit. Recently, the scaling limit of near critical percolation was also constructed by Garban, Pete and Schramm. The aim of this article is to explain how these results imply the convergence, as $p$ tends to $p_c$, of the Wulff crystal to a Euclidean disk. The main ingredient of the proof is the rotational invariance of the scaling limit of near-critical percolation proved by these three mathematicians.

Highlights

Definition of the model Percolation as a physical model was introduced by Broadbent and Hammersley in the fifties [5]
The scaling limit of near-critical percolation was constructed by Garban, Pete and Schramm
The main ingredient of the proof is the rotational invariance of the scaling limit of near-critical percolation proved by these three mathematicians

Summary

Introduction

Definition of the model Percolation as a physical model was introduced by Broadbent and Hammersley in the fifties [5]. For percolation on the triangular lattice, τp(u)/τp(|u|) −→ 1 uniformly in the direction u ∈ U as p pc While this result is very intuitive once conformal invariance has been proved, it does not follow directly from it. Other models Let us mention that conformal invariance has been proved for a number of models, including the dimer model [23] and the Ising model [30, 15]; see [17] for lecture notes on the subject In both cases, exact computations (see [26] for the Ising model) allow one to show that the inverse correlation length becomes isotropic, providing an extension of Theorem 1.1. For two sets A and B in R2, we say that A ←→ B if there exists a ∈ A ∩ T and b ∈ B ∩ T such that a ←→ b

An important input

Proof of the theorem