Title: Some remarks on <i>AB</i>-percolation in high dimensions
Abstract:In this paper we consider the AB-percolation model on Z+d and Zd. Let pHalt(Zd) be the critical probability for AB-percolation on Zd. We show that pHalt(Zd)∼1/(2d2). If the probability of a site to be...In this paper we consider the AB-percolation model on Z+d and Zd. Let pHalt(Zd) be the critical probability for AB-percolation on Zd. We show that pHalt(Zd)∼1/(2d2). If the probability of a site to be in state A is γ/(2d2) for some fixed γ&gt;1, then the probability that AB-percolation occurs converges as d→∞ to the unique strictly positive solution y(γ) of the equation y=1−exp(−γy). We also find the limit for the analogous quantities for oriented AB-percolation on Z+d. In particular, pHalt(Z+d)∼2/d2. We further obtain a small extension to the two parameter problem in which even vertices of Zd have probability pA of being in state A and odd vertices have probability pB of being in state B (but without relation between pA and pB). The principal tools in the proofs are a method of Penrose (1993) for asymptotics of percolation on graphs with vertices of high degree and the second moment method.Read More
Publication Year: 2000
Publication Date: 2000-03-01
Language: en
Type: article
Indexed In: ['crossref']
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Cited By Count: 2
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