Abstract.
We consider oriented bond or site percolation on ℤd +. In the case of bond percolation we denote by P p the probability measure on configurations of open and closed bonds which makes all bonds of ℤd + independent, and for which P p {e is open} = 1 −P p e {is closed} = p for each fixed edge e of ℤd +. We take X(e) = 1 (0) if e is open (respectively, closed). We say that ρ-percolation occurs for some given 0 < ρ≤ 1, if there exists an oriented infinite path v 0 = 0, v 1, v 2, …, starting at the origin, such that lim inf n →∞ (1/n) ∑ i=1 n X(e i ) ≥ρ, where e i is the edge {v i−1 , v i }. [MZ92] showed that there exists a critical probability p c = p c (ρ, d) = p c (ρ, d, bond) such that there is a.s. no ρ-percolation for p < p c and that P p {ρ-percolation occurs} > 0 for p > p c . Here we find lim d →∞ d 1/ρ p c (ρd, bond) = D 1 , say. We also find the limit for the analogous quantity for site percolation, that is D 2 = lim d →∞ d 1/ρ p c (ρ, d, site). It turns out that for ρ < 1, D 1 < D 2 , and neither of these limits equals the analogous limit for the regular d-ary trees.
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Received: 7 January 1999 / Published online: 14 June 2000
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Kesten, H., Su, ZG. Asymptotic behavior of the critical probability for ρ-percolation in high dimensions. Probab Theory Relat Fields 117, 419–447 (2000). https://doi.org/10.1007/s004400050013
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DOI: https://doi.org/10.1007/s004400050013
- Mathematics Subject Classification (1991): Primary 60K35; Secondary 82B43
- Key words and phrases:ρ-percolation – Critical probability – Second moment method