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

Let p c (d) be the critical probability for oriented percolation in ℤ d 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 → ∞, dp c (d) and d μ,( d ) → γ where γ is a constant in [e −1 , 2 −1 ] which we conjecture to be e −1 . In the case of p c (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):