Existence and continuity of the flow constant in first passage percolation

Abstract

We consider the model of i.i.d. first passage percolation on Z d , where we associate with the edges of the graph a family of i.i.d. random variables with common distribution G on [0, +∞] (including +∞). Whereas the time constant is associated to the study of 1-dimensional paths with minimal weight, namely geodesics, the flow constant is associated to the study of (d−1)-dimensional surfaces with minimal weight. In this article, we investigate the existence of the flow constant under the only hypothesis that G({+∞}) < p c (d) (in particular without any moment assumption), the convergence of some natural maximal flows towards this constant, and the continuity of this constant with regard to the distribution G.• the node law: for every vertex x in Ω (G 1 ∩ G 2 ), we have y∈Z d : e= x,y ∈E d ∩Ω f (e) 2 1 f (e)/ f (e) 2 = xy − 1 f (e)/ f (e) 2 = yx = 0 , i.e., there is no loss of fluid inside Ω;• the capacity constraint: for every edge e in Ω, we havei.e., the amount of water that flows through e per second cannot exceed its capacity t G (e).

Size of a minimal cutset in supercritical first passage percolation

“…The maximal flow properly renormalized converges towards the so-called flow constant. In [16], Rossignol and Théret proved the same results without any moment assumption on G, they even allow the capacities to take infinite value as long as G({+∞}) < p c (d) where p c (d) denotes the critical parameter of i.i.d. bond percolation on Z d .…”

“…In [16], Rossignol and Théret extended the previous results without any moment condition on G, they even allow G to have an atom in +∞ as long as G({+∞}) < p c (d). They proved the following law of large numbers for the maximal flow from the top to the bottom of flat cylinders.…”

“…Remark 1. Note that here we will work only with the cylinder cyl(A, h) whereas the authors of [16] work mainly with the cylinder cyl (A, h).…”

The maximal flow from a compact convex subset to infinity in first passage percolation on $\mathbb{Z}^{d}$

“…The reduced boundary ∂ * S of S is a subset of ∂S such that, at each point x of ∂ * S, it is possible to define a normal vector n S (x) to S in a measure-theoretic sense, and moreover P(S) = H d−1 (∂ * S). We denote by ν the flow constant that is a function from the unit sphere S d−1 of R d to R + as defined in [12]. We denote by ν max and ν min its maximal and minimal values on the sphere.…”

“…This question was addressed in [10], [11] and [14] where one can find laws of large numbers and large deviation estimates for this maximal flow when the dimensions of the box grow to infinity under some moments assumptions on the capacities and on the direction v. The maximal flow properly renormalized converges towards the so-called flow constant ν(v). In [12], Rossignol and Théret proved the same results without any moment assumption on G for any direction v. Roughly speaking, the flow constant ν(v) corresponds to the expected maximal amount of water that can flow per second in the direction of v. Let us consider a point x in ∂A with its associated normal unit exterior vector n A (x) and infinitesimal surface S(x) around x. When we consider nA, an enlarged version of A, the surface S(x) becomes nS(x) and the expected maximal amount of water that can flow in the box of basis nS(x) in the direction n A (x) is of order n d−1 ν(n A (x))C S where C S is a constant depending on the area of the surface.…”