Continuity of the time and isoperimetric constants in supercritical percolation

Abstract

We consider two different objects on supercritical Bernoulli percolation on the edges of Z d : the time constant for i.i.d. first-passage percolation (for d ≥ 2) and the isoperimetric constant (for d = 2). We prove that both objects are continuous with respect to the law of the environment. More precisely we prove that the isoperimetric constant of supercritical percolation in Z 2 is continuous in the percolation parameter. As a corollary we obtain that normalized sets achieving the isoperimetric constant are continuous with respect to the Hausdroff metric. Concerning first-passage percolation, equivalently we consider the model of i.i.d. first-passage percolation on Z d with possibly infinite passage times: we associate with each edge e of the graph a passage time t(e) taking values in [0, +∞], such that P[t(e) < +∞] > pc (d).We prove the continuity of the time constant with respect to the law of the passage times. This extends the continuity property of the asymptotic shape previously proved by Cox and Kesten [8,10,19] for first-passage percolation with finite passage times. 2010 Mathematics Subject Classification. 60K35, 82B43. arXiv:1512.00742v2 [math.PR] 30 May 2016 1.2. First-passage percolation on the infinite cluster in dimension d ≥ 2. Consider a fixed dimension d ≥ 2. First-passage percolation on Z d was introduced by Hammersley and Welsh [16] as a model for the spread of a fluid in a porous medium. To each edge of the Z d lattice is attached a nonnegative random variable t(e) which corresponds to the travel time needed by the fluid to cross the edge. When the passage times are independent identically distributed variables with common distribution G, with suitable moment conditions, the time needed to travel from 0 to nx is equivalent to nµ G (x), where µ G is a semi-norm associated to G called the time constant; Cox and Durrett [9] proved this result under necessary and sufficient integrability conditions on the distribution G of the passage times. Kesten in [17] proved that the semi-norm µ G is a norm if and only if G({0}) < p c (d).In casual terms, the asymptotic shape theorem (in its geometric form) says that in this case, the random ball of radius n, i.e. the set of points that can be reached within time n from the origin, asymptotically looks like nB µ G , where B µ G is the unit ball for the norm µ G . The ball B µ G is thus called the asymptotic shape associated to G.A natural extension is to replace the Z d lattice by a random environment given by the infinite cluster C ∞ of a supercritical Bernoulli percolation model. This is equivalent to allow t(e) to be equal to +∞. The existence of a time constant in firstpassage percolation in this setting was first proved by Garet and Marchand in [12], in the case where (t(e)1 1 t(e)<+∞ ) is a stationary integrable ergodic field. Recently, Cerf and Théret [6] focused of the case where (t(e)1 1 t(e)<+∞ ) is an independent field, and managed to prove the existence of an appropriate time constant without any integrability assumption. In the following, we ad...

Anchored isoperimetric profile of the infinite cluster in supercritical bond percolation is Lipschitz continuous

“…In this paper, we aim to study the regularity properties of the anchored isoperimetric profile. This was first studied by Garet, Marchand, Procaccia, Théret in [5], they proved that the modified Cheeger constant in dimension 2 is continuous on (p c (2), 1]. The aim of this paper is the proof of the two following theorems.…”

Existence and continuity of the flow constant in first passage percolation

“…Moreover, the technology used to prove the key Lemma 2.9 (using the contours) is directly inspired by the study of the existence of the time constant without any moment condition. The proof of the continuity of the flow constant, Theorem 2.6, we propose in this paper is heavily influenced by the proofs of the continuity of the time constant given in [12,22,15]. The real difficulty of our work is to extend the definition of the flow constant to probability measure with infinite mean -once this is done, it is harmless to admit probability measures F on [0, +∞] such that F ({+∞}) < p c (d), we do not even have to use a renormalization argument.…”

“…This section is devoted to the proof of Theorem 2.6. To prove this theorem we mimick the proof of the corresponding property for the time constant, see [10], [12], [22] and [15]. We stress the fact that the proof relies heavily on these facts:…”

Book Review: 50 years of first-passage percolation

“…(1) The limiting shape and the function μ are known to be continuous with respect to weak convergence; see [10,15,18]. That is, if F n ⇒ F and if μ n , μ are respective limits, then lim n→∞ μ n (x) = μ(x) for every x ∈ R d .…”