We study isoperimetric sets, i.e., sets with minimal boundary for a prescribed volume, on the unique infinite connected component of supercritical bond percolation on the square lattice. In the limit of the volume tending to infinity, properly scaled isoperimetric sets are shown to converge (in the Hausdorff metric) to the solution of an isoperimetric problem in R 2 with respect to a particular norm. As part of the proof we also show that the anchored isoperimetric profile as well as the Cheeger constant of the giant component in finite boxes scale to deterministic quantities. This settles a conjecture of Itai Benjamini for the square lattice.

We prove a quenched invariance principle for simple random walk on the unique infinite percolation cluster for a general class of percolation models on Z d , d ≥ 2, with long-range correlations introduced in (Drewitz et al. in J Math Phys 55(8):083307, 2014), solving one of the open problems from there. This gives new results for random interlacements in dimension d ≥ 3 at every level, as well as for the vacant set of random interlacements and the level sets of the Gaussian free field in the regime of the so-called local uniqueness (which is believed to coincide with the whole supercritical regime). An essential ingredient of our proof is a new isoperimetric inequality for correlated percolation models. Mathematics Subject Classification

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...

We consider the geometry of random interlacements on the d-dimensional lattice. We use ideas from stochastic dimension theory developed in [BKPS04] to prove the following: Given that two vertices x, y belong to the interlacement set, it is possible to find a path between x and y contained in the trace left by at most ⌈d/2⌉ trajectories from the underlying Poisson point process. Moreover, this result is sharp in the sense that there are pairs of points in the interlacement set which cannot be connected by a path using the traces of at most ⌈d/2⌉ − 1 trajectories.

We consider two stationary versions of the Eden model, on the upper half planar lattice, resulting in an infinite forest covering the half plane. Under weak assumptions on the weight distribution and by relying on ergodic theorems, we prove that almost surely all trees are finite. Using the mass transport principle, we generalize the result to Eden model in graphs of the form G × Z+, where G is a Cayley graph. This generalizes certain known results on the two-type Richardson model, in particular of Deijfen and Häggström in 2007 [5].