Asymptotic behavior of the critical probability for ρ-percolation in high dimensions

We consider oriented bond or site percolation on Z(+)(d). In the case of bond per colation we denote by P-p the probability measure on configurations of open and closed bonds which makes all bonds of Z(+)(d) independent, and for which P-p{e is open} = 1 - P-p{e is closed} = p for each fixed edge e of Z(+)(d). We take X(e) = 1(0) if e is open (respectively, closed). We say that rho-percolation occurs for some given 0 < rho less than or equal to 1, if there exists an oriented infinite path v(0) = 0, v(1), v(2),..., starting at the origin, such that lim inf(n-->infinity) (1/n) Sigma(1=i)(n) X(e(i)) greater than or equal to rho, where e(i) is the edge {v(i-1), v(i)}. [MZ92] showed that there exists a critical probability p(c) = p(c)(rho, d) = p(c)(rho, d, bond) such that there is a.s. no rho-percolation for p < p(c) and that P-p{rho-percolation occurs} > 0 for p > p(c). Here we find lim(d-->infinity) d(1/rho) p(c)(rho, d, bond) = D-1, say. We also find the limit for the analogous quantity for site percolation, that is D-2 = lim(d-->infinity) d(1/rho) p(c)(rho, d, site). It turns out that for rho < 1, D-1 < D-2, and neither of these limits equals the analogous limit for the regular d-ary trees.