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Monotonicity in first-passage percolation. (English) Zbl 1301.60108

Summary: We consider standard first-passage percolation on \(\mathbb Z^d\). Let \(e_1\) be the first coordinate vector. Let \(a(n)\) be the expected passage time from the origin to \(ne_1\). In this short paper, we note that \(a(n)\) is increasing under some strong condition on the support of the distribution of the passage times on the edges.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
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