Oriented percolation in dimensions d ≥ 4: bounds and asymptotic formulas

Let pc(d) be the critical probability for oriented percolation in ℤd and let μ(d) be the time constant for the first passage process based on the exponential distribution. In this paper we show that as d → ∞,dpc(d) and dμ,(d) → γ where γ is a constant in [e−1, 2−1] which we conjecture to be e−1. In the case of pc(d) we have made some progress toward obtaining an asymptotic expansion in powers of d−1. Our results show The left hand side agrees, up to O(d−3), with a (nonrigorous) series expansion of Blease (1, 2):