Vanishing of the anchored isoperimetric profile in bond percolation at p c

Abstract

We consider the anchored isoperimetric profile of the infinite open cluster, defined for $p>p\_c$, whose existence has been recently proved in [3]. We extend adequately the definition for $p=p\_c$, in finite boxes. We prove a partial result which implies that, if the limit defining the anchored isoperimetric profile at $p\_c$ exists, it has to vanish.