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

Raphaël Cerf (1, 2) , Barbara Dembin (3)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 DMA - Département de Mathématiques et Applications - ENS Paris
3 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation

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.

Domaines: Probabilités [math.PR]

Raphaël Cerf, Barbara Dembin. Vanishing of the anchored isoperimetric profile in bond percolation at p c. 2019. ⟨hal-02071030v2⟩