TY - JOUR

T1 - Oriented Percolation in Dimensions D ≥ 4

T2 - Bounds and Asymptotic Formulas

AU - Cox, J. Theodore

AU - Durrett, Richard

N1 - Funding Information:
It is a pleasure to thank Harry Kesten for his help with this work and for allowing us to present his proof ofpc(d) ^ p{d). J. T. Cox was partially supported by NSF grant MCS 81-02131. This work was began while R. Durrett was visiting Cornell University and partially supported by an NSF grant to that University.

PY - 1983/1

Y1 - 1983/1

N2 - Let pc(d) be the critical probability for oriented percolation in Zdand let (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)→ 1 and dμ{d)→γ where y 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).

AB - Let pc(d) be the critical probability for oriented percolation in Zdand let (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)→ 1 and dμ{d)→γ where y 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).

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U2 - 10.1017/S0305004100060436

DO - 10.1017/S0305004100060436

M3 - Article

AN - SCOPUS:84976077801

VL - 93

SP - 151

EP - 162

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

SN - 0305-0041

IS - 1

ER -