Lower large deviations for the maximal flow through a domain of $${\mathbb{R}^d}$$ in first passage percolation

Abstract

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

Law of large numbers for the maximal flow through a domain of $\mathbb{R}^{d}$ in first passage percolation

“…In the two companion papers [7] and [8], we prove in fact that the lower large deviations of φ n /n d−1 below φ Ω are of surface order and that the upper large deviations of φ n /n d−1 above φ Ω are of volume order (see section 3.2 where these results are presented). If X is a subset of R d included in a hyperplane of R d and of codimension 1 (for example a non-degenerate hyperrectangle), we denote by hyp(X) the hyperplane spanned by X, and we denote by cyl(X, h) the cylinder of basis X and of height 2h defined by cyl(X, h) = {x + tv | x ∈ X , t ∈ [−h, h]} , where v is one of the two unit vectors orthogonal to hyp(X) (see Figure 2).…”

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

“…Step (iii): The remaining of the proof follows the same ideas as in [8]. We link the probability defined in the right hand side of (55) with the probability that the flow is abnormally small in some local region of ∂F ∩ Ω .…”

“…The last thing to do is then to cover the disc(x, r, v) by hyperrectangles in order to use the estimate that the flow is abnormally small in a cylinder. This work was done in section 6 in [8]. It is possible to choose δ 2 depending on F 1 , .…”

“…Next, we choose δ such that η(δ) satisfies inequalities (36), (50) and (52). We choose δ 2 depending on w (and so on ε) and G. The parameter δ 2 has to satisfy some inequalities that we do not detail here, we refer to section 7 in [8]. Finally, to each r in {1, .…”

“…It is possible to choose δ 2 depending on W , G and w (see again section 6 in [8]) such that there exist positive constants C 1,k and C 2,k depending on G, d, W , k and w so that for all k ∈ J, P[G(x k , ρ k , n Wp (x k ), w, δ 2 )] ≤ C 1,k exp(−C 2,k n d−1 ) .…”