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

  • J. Theodore Cox (a1) and Richard Durrett (a1)

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):

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References

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(1)Blease, J.Directed bond percolation on hypercubical lattices. J. Phys. C. 10 (1977), 925936.
(2)Blease, J.Pair connectedness for directed bond percolation on some 2d lattices by series methods. J. Phys. C 10 (1977), 34613476.
(3)Cox, T. and Durrett, R.Some limit theorems for percolation processes with necessary and sufficient conditions. Ann. Probability 9 (1981), 583603.
(4)Feller, W.An introduction to probability theory and its applications, vol. I, third edition (John Wiley and Sons, New York, 1970).
(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, 1737.
(6)Mityugin, L. G.Some multidimensional systems of automata related to percolation problems. Problems of Information Transmission 11 (1975), 259261.
(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).
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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