Vanishing of the anchored isoperimetric profile in bond percolation at $p_{c}$

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

“…The goal of an isoperimetric problem is to characterize sets of a prescribed volume with minimal boundary measure. The classical isoperimetric problems are stated in the continuum and have a long history, but they have recently been studied in the discrete setting as well (see, e.g., [5,9,12,13,15,16,17,23] for such studies in bond percolation on Z d ). The Cheeger constant is a way of encoding such problems for graphs.…”