Mathematics – Probability
Scientific paper
2009-07-31
Mathematics
Probability
41 pages, 8 figures
Scientific paper
We consider the standard first passage percolation model in the rescaled graph $\mathbb{Z}^d/n$ for $d\geq 2$, 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. Moreover, we give a necessary and sufficient condition on the law of the capacity of the edges to ensure that $\phi_{\Omega} >0$.
Cerf Raphaël
Théret Marie
No associations
LandOfFree
Law of large numbers for the maximal flow through a domain of $\mathbb{R}^d$ in first passage percolation does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Law of large numbers for the maximal flow through a domain of $\mathbb{R}^d$ in first passage percolation, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Law of large numbers for the maximal flow through a domain of $\mathbb{R}^d$ in first passage percolation will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-267560