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

Published online by Cambridge University Press:  24 October 2008

J. Theodore Cox  and
Richard Durrett
J. Theodore Cox
Affiliation:
Syracuse University and U.C.L.A.
Richard Durrett
Affiliation:
Syracuse University and U.C.L.A.
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Abstract

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):

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 93 , Issue 1 , January 1983 , pp. 151 - 162
Copyright
Copyright © Cambridge Philosophical Society 1983

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References

REFERENCES

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