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We consider the standard first passage percolation model in the rescaled graph Z d /n for d ≥ 2, and a domain of boundary in R d . Let 1 and 2 be two disjoint open subsets of , representing the parts of through which some water can enter and escape from . We investigate the asymptotic behaviour of the flow φ n through a discrete version n of between the corresponding discrete sets 1 n and 2 n . We prove that under some conditions on the regularity of the domain and on the law of the capacity of the edges, the lower large deviations of φ n /n d−1 below a certain constant are of surface order.Keywords First passage percolation · Maximal flow · Minimal cut · Large deviations Mathematics Subject Classification (2000) 60K35 First definitions and main resultWe use many notations introduced in [8,9]. Let d ≥ 2. We consider the graph (Z d n , E d n ) having for vertices Z d n = Z d /n and for edges E d n , the set of pairs of nearest neighbours for the standard L 1 norm. With each edge e in E d n we associate a random variable t (e) with values in R + . We suppose that the family (t (e), e ∈ E d n ) is independent and identically distributed, with a common law : this is the standard model of first passage

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 model of i.i.d. first passage percolation on Z d : we associate with each edge e of the graph a passage time t(e) taking values in [0, +∞], such that P[t(e) < +∞] > p c (d). Equivalently, we consider a standard (finite) i.i.d. first passage percolation model on a super-critical Bernoulli percolation performed independently. We prove a weak shape theorem without any moment assumption. We also prove that the corresponding time constant is positive if and only ifWe consider a family of i.i.d. random variables (t(e), e ∈ E d ) associated to the edges of the graph, taking values in [0, +∞] (we emphasize that +∞ is included here). We denote by F the common distribution of these variables. We interpret t(e) as the time needed to cross the edge e. If x, y are vertices in Z d , a path r from x to y is a sequence r = (v 0 , e 1 , v 1 , . . . , e n , v n ) of vertices (v i , i = 0, . . . , n) and edges (e i , i = 1, . . . , n) for some n ∈ N such that v 0 = x, v n = y and for all i ∈ {1, . . . , n}, e i = v i−1 , v i . For any path r, we define T (r) = e∈r t(e). We obtain a random pseudo-metric T on Z d in the following way : ∀x, y ∈ Z d , T (x, y) = inf{T (r) | r is a path from x to y} .

We consider the standard first passage percolation model in the rescaled graph Z d /n for d ≥ 2 and a domain Ω of boundary Γ in R d . Let Γ 1 and Γ 2 be two disjoint open subsets of Γ representing the parts of Γ through which some water can enter and escape from Ω. We investigate the asymptotic behavior of the flow φn through a discrete version Ωn of Ω between the corresponding discrete sets Γ 1 n and Γ 2 n . We prove that under some conditions on the regularity of the domain and on the law of the capacity of the edges, the upper large deviations of φn/n d−1 above a certain constant are of volume order, that is, decays exponentially fast with n d . This article is part of a larger project in which the authors prove that this constant is the a.s. limit of φn/n d−1 .