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We consider isoperimetric sets, i.e., sets with minimal vertex boundary for aprescribed volume, of the infinite cluster of supercritical site percolation onthe triangular lattice. Let $p$ be the percolation parameter and let $p_c$ bethe critical point. By adapting the proof of Biskup, Louidor, Procaccia andRosenthal [6] for isoperimetry in bond percolation on the square lattice, weshow that the isoperimetric sets, when suitably rescaled, converge almostsurely to a translation of the normalized Wulff crystal $\widehat{W}_p$. Moreimportantly, 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 fornear-critical Bernoulli first-passage percolation proved by the authorrecently.

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