Existence and continuity of the flow constant in first passage percolation

Raphaël Rossignol (1) , Marie Théret (2)

1 IF - Institut Fourier
2 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation

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.

Mots clés: continuity; First passage percolation; maximal flow; minimal cutset

Domaines: Probabilités [math.PR]

Raphaël Rossignol, Marie Théret. Existence and continuity of the flow constant in first passage percolation. Electronic Journal of Probability, 2018, 23 (99), ⟨10.1214/18-ejp214⟩. ⟨hal-01568243v2⟩