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Law of large numbers for the maximal flow through a domain of \(\mathbb R^d\) in first passage percolation. (English) Zbl 1228.60107

The paper deals with the standard first passage percolation model in the rescaled graph \(\mathbb R^d/n\) for \(d\geq 1\), and domain \( \Omega\) of boundary \(\Gamma\) in \(\mathbb R^d\). Let \(\Gamma^1\) and \(\Gamma^2\) be two disjoint open subsets of \(\Gamma\) through which some water can enter and escape from \(\Omega\). The asymptotic behaviour of the flow \(\varphi_n\) through a discrete version \(\Omega_n\) of \(\Omega\) between the corresponding discrete sets \(\Gamma^1\) and \(\Gamma^2\) is investigated. It is proved that, under some conditions on the regularity of the domain and on the law of the capacity of the edges, \(\varphi_n\) converges almost surely towards a constant \(\varphi_\Omega\), which is the solution of a continuous non-random min-cut problem. In addition, the authors give a necessary and sufficient condition on the law of the capacity of the edges to ensure that \(\varphi_\Omega>0\).

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
49Q20 Variational problems in a geometric measure-theoretic setting
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