Limit of the Wulff crystal when approaching criticality for isoperimetry in 2D percolation

Abstract

We consider isoperimetric sets, i.e., sets with minimal vertex boundary for a prescribed volume, of the infinite cluster of supercritical site percolation on the triangular lattice. Let $p$ be the percolation parameter and let $p_c$ be the critical point. By adapting the proof of Biskup, Louidor, Procaccia and Rosenthal [6] for isoperimetry in bond percolation on the square lattice, we show that the isoperimetric sets, when suitably rescaled, converge almost surely to a translation of the normalized Wulff crystal $\widehat{W}_p$. More importantly, we prove that $\widehat{W}_p$ tends to a Euclidean disk as $p\downarrow p_c$. This settles the site version of a conjecture proposed in [6]. A key input to the proof is the convergence of the limit shapes for near-critical Bernoulli first-passage percolation proved by the author recently.