## Electronic Journal of Probability

### Existence and continuity of the flow constant in first passage percolation

#### Abstract

We consider the model of i.i.d. first passage percolation on $\mathbb{Z} ^d$, where we associate with the edges of the graph a family of i.i.d. random variables with common distribution $G$ on $[0,+\infty ]$ (including $+\infty$). 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 $G(\{+\infty \} ) < p_c(d)$ (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 $G$.

#### Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 99, 42 pp.

Dates
Accepted: 20 August 2018
First available in Project Euclid: 26 September 2018

https://projecteuclid.org/euclid.ejp/1537927580

Digital Object Identifier
doi:10.1214/18-EJP214

Mathematical Reviews number (MathSciNet)
MR3862614

Zentralblatt MATH identifier
06964793

#### Citation

Rossignol, Raphaël; Théret, Marie. Existence and continuity of the flow constant in first passage percolation. Electron. J. Probab. 23 (2018), paper no. 99, 42 pp. doi:10.1214/18-EJP214. https://projecteuclid.org/euclid.ejp/1537927580

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