Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-23T07:32:11.576Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Blease, J.Directed bond percolation on hypercubical lattices. J. Phys. C. 10 (1977), 925–936.Google Scholar
(2)Blease, J.Pair connectedness for directed bond percolation on some 2d lattices by series methods. J. Phys. C 10 (1977), 3461–3476.CrossRefGoogle Scholar
(3)Cox, T. and Durrett, R.Some limit theorems for percolation processes with necessary and sufficient conditions. Ann. Probability 9 (1981), 583–603.Google Scholar
(4)Feller, W.An introduction to probability theory and its applications, vol. I, third edition (John Wiley and Sons, New York, 1970).Google Scholar
(5)Hammersley, J. Bornes supérieures de la probabilité critique dans un processus de filtration. Colloques internationaux du Centre National de la Recherche Scientifique, 87 (1959), Paris, 15–20 juillet 1958, 17–37.Google Scholar
(6)Mityugin, L. G.Some multidimensional systems of automata related to percolation problems. Problems of Information Transmission 11 (1975), 259–261.Google Scholar
(7)Smythe, R. T. and Wierman, J. C. First-passage percolation on the square lattice. Lecture Notes in Mathematics, no. 671 (Springer-Verlag, New York, 1978).Google Scholar