The maximal flow through a domain of $mathbb{R}^d$ in first passage percolation


Il Prof. Raphael Cerf, dell'Universite' de Paris-Sud, Orsay, terra'
lunedi 26 aprile alle ore 14:00 nell'Aula 2AB45 della Torre Archimede, un seminario dal titolo "The maximal flow through a domain of $mathbb{R}^d$ in first passage percolation" joint work with Marie Theret".

We consider the standard first passage percolation model in the
rescaled graph $mathbb{Z}^d/n$ for $dgeq 2$, and a domain $Omega$
of boundary $Gamma$ in $mathbb{R}^d$. Let $Gamma^1$ and $Gamma^2$
be two disjoint open subsets of $Gamma$, representing the parts of
$Gamma$ through which some water can enter and escape from $Omega$.
We investigate the asymptotic behaviour of the flow $phi_n$ through a
discrete version $Omega_n$ of $Omega$ between the corresponding
discrete sets $Gamma^1_n$ and $Gamma^2_n$. We prove that under some
conditions on the regularity of the domain and on the law of the
capacity of the edges, $phi_n$ converges almost surely towards a
constant $phi_{Omega}$, which is the solution of a continuous
non-random min-cut problem.

Rif. int. P. Dai Pra