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

TL;DR

This paper explains how recent advances in the understanding of near-critical percolation on the triangular lattice imply that the Wulff crystal converges to a Euclidean disk as the percolation parameter approaches criticality, highlighting the role of rotational invariance.

Contribution

It demonstrates the convergence of the Wulff crystal to a Euclidean disk near criticality using recent results on the scaling limit and rotational invariance of near-critical percolation.

Findings

Wulff crystal converges to a Euclidean disk as p approaches p_c

Rotational invariance of the scaling limit is key to the proof

Recent results on near-critical percolation enable this convergence

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 very modest contribution 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.