Law of large numbers for the maximal flow through a domain of $\mathbb{R}^d$ in first passage percolation

Mathematics – Probability

Scientific paper

Rate now

  [ 0.00 ] – not rated yet Voters 0   Comments 0

Details

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 $\phi_{\Omega} >0$.

No associations

LandOfFree

Say what you really think

Search LandOfFree.com for scientists and scientific papers. Rate them and share your experience with other people.

Rating

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.

Rate now

     

Profile ID: LFWR-SCP-O-267560

  Search
All data on this website is collected from public sources. Our data reflects the most accurate information available at the time of publication.