Existence of the anchored isoperimetric profile in supercritical bond percolation in dimension two and higher

Barbara Dembin (1)

1 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation

Let d ≥ 2. We consider an i.i.d. supercritical bond percolation on Z^d , every edge is open with a probability p > p_c (d), where p_c (d) denotes the critical point. We condition on the event that 0 belongs to the infinite cluster C_∞ and we consider connected subgraphs of C_∞ having at most n^d vertices and containing 0. Among these subgraphs, we are interested in the ones that minimize the open edge boundary size to volume ratio. These minimizers properly rescaled converge towards a translate of a deterministic shape and their open edge boundary size to volume ratio properly rescaled converges towards a deterministic constant.

Mots clés: Anchored isoperimetric profile; Percolation

Domaines: Probabilités [math.PR]

Barbara Dembin. Existence of the anchored isoperimetric profile in supercritical bond percolation in dimension two and higher. ALEA : Latin American Journal of Probability and Mathematical Statistics, 2020, 17 (1), pp.205. ⟨10.30757/alea.v17-09⟩. ⟨hal-01905034⟩