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Line-of-sight percolation (2008)
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@MISC{Bollobás08line-of-sightpercolation,
author = {Béla Bollobás and Svante Janson and Oliver Riordan},
title = {Line-of-sight percolation},
year = {2008}
}
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Abstract
Given ω ≥ 1, let Z 2 (ω) be the graph with vertex set Z 2 in which two vertices are joined if they agree in one coordinate and differ by at most ω in the other. (Thus Z 2 (1) is precisely Z 2.) Let pc(ω) be the critical probability for site percolation on Z 2 (ω). Extending recent results of Frieze, Kleinberg, Ravi and Debany, we show that limω→ ∞ ωpc(ω) = log(3/2). We also prove analogues of this result for the n-by-n grid and in higher dimensions, the latter involving interesting connections to Gilbert’s continuum percolation model. To prove our results, we explore the component of the origin in a certain non-standard way, and show that this exploration is well approximated by a certain branching random walk.
