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.

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 parameter for this percolation. We know that there exists almost surely a unique infinite open cluster C p [11]. We are interested in the regularity properties of the chemical distance for supercritical Bernoulli percolation. The chemical distance between two points x, y ∈ C p corresponds to the length of the shortest path in C p joining the two points. The chemical distance between 0 and nx grows asymptotically like nµ p (x). We aim to study the regularity properties of the map p → µ p in the supercritical regime. This may be seen as a special case of first passage percolation where the distribution of the passage time isIt is already known that the map p → µ p is continuous (see [10]).

We consider the standard first passage percolation model in the rescaled lattice Z d /n for d ≥ 2 and a bounded domain Ω in R d . We denote by Γ 1 and Γ 2 two disjoint subsets of ∂Ω representing respectively the source and the sink, i.e., where the water can enter in Ω and escape from Ω. A maximal stream is a vector measure − → µ max n that describes how the maximal amount of fluid can enter through Γ 1 and spreads in Ω. Under some assumptions on Ω and G, we already know a law of large number for − → µ max n . The sequence ( − → µ max n ) n≥1 converges almost surely to the set of solutions of a continuous deterministic problem of maximal stream in an anisotropic network. We aim here to derive a large deviation principle for streams and deduce by contraction principle the existence of a rate function for the upper large deviations of the maximal flow in Ω.

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 parameter for this percolation. We know that there exists almost surely a unique infinite open cluster C_p [11]. We are interested in the regularity properties of the chemical distance for supercritical Bernoulli percolation. The chemical distance between two points x, y ∈ C_p corresponds to the length of the shortest path in C_p joining the two points. The chemical distance between 0 and nx grows asymptotically like nµ_p(x). We aim to study the regularity properties of the map p → µ_p in the supercritical regime. This may be seen as a special case of first passage percolation where the distribution of the passage time is G_p = pδ_1 + (1 − p)δ_∞, p > p_c(d). It is already known that the map p → µ_p is continuous (see [10]).

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 parameter for this percolation. We know that there exists almost surely a unique infinite open cluster C p [7]. We are interested in the regularity properties in p of the anchored isoperimetric profile of the infinite cluster C p . For d ≥ 2, we prove that the anchored isoperimetric profile defined in [4] is Lipschitz continuous on all intervals [p 0 , p 1 ] ⊂ (p c (d), 1).