Some remarks on AB-percolation in high dimensions

In this paper we consider the AB-percolation model on Z1 d and Z. Let pH (Zd) be the critical probability for AB-percolation on Z. We show that pH (Zd) ;1/(2d). If the probability of a site to be in state A is g/(2d) for some fixed g.1, then the probability that AB-percolation occurs converges as d→` to the unique strictly positive solution y(g) of the equation y512exp(2gy). We also find the limit for the analogous quantities for oriented AB-percolation on Z1 d . In particular, pH (Z1 d );2/d. We further obtain a small extension to the two parameter problem in which even vertices of Z have probability p of being in state A and odd vertices have probability p of being in state B ~but without relation between p and p!. 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. © 2000 American Institute of Physics. @S0022-2488~00!01303-7#