The Annals of Probability

Unpredictable paths and percolation

Itai Benjamini, Robin Pemantle, and Yuval Peres

Full-text: Open access

Abstract

4 We construct a nearest-neighbor process ${S_n}$ on Z that is less predictable than simple random walk, in the sense that given the process until time $n$, the conditional probability that $S_{n+k} = x$ is uniformly bounded by $Ck^{-\infty}$ for some $\alpha > 1/2$. From this process, we obtain a probability measure $\mu$ on oriented paths in $\mathbf{Z}^3$ such that the number of intersections of two paths, chosen independently according to $\mu$, has an exponential tail. (For $d \geq 4$, the uniform measure on oriented paths from the origin in $\mathbf{Z}^d$ has this property.) We show that on any graph where such a measure on paths exists, oriented percolation clusters are transient if the retention parameter $p$ is close enough to 1. This yields an extension of a theorem of Grimmett, Kesten and Zhang, who proved that supercritical percolation clusters in $\mathbf{Z}^d$ are transient for all $d \geq 3$.

Article information

Source
Ann. Probab., Volume 26, Number 3 (1998), 1198-1211.

Dates
First available in Project Euclid: 31 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022855749

Digital Object Identifier
doi:10.1214/aop/1022855749

Mathematical Reviews number (MathSciNet)
MR1634419

Zentralblatt MATH identifier
0937.60070

Subjects
Primary: 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60J65: Brownian motion [See also 58J65] 60J15 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Percolation transience electrical networks multitype branching process

Citation

Benjamini, Itai; Pemantle, Robin; Peres, Yuval. Unpredictable paths and percolation. Ann. Probab. 26 (1998), no. 3, 1198--1211. doi:10.1214/aop/1022855749. https://projecteuclid.org/euclid.aop/1022855749


Export citation

References

  • ANTAL, P. and PISZTORA, A. 1996. On the chemical distance in supercritical Bernoulli percolation. Ann. Probab. 24 1036 1048.
  • ATHREYA, K. and NEY, P. 1972. Branching Processes. Springer, New York.
  • COX, T. and DURRETT, R. 1983. Oriented percolation in dimensions d 4; bounds and asymptotic formulas. Math. Proc. Cambridge Philos. Soc. 93 151 162.
  • DOYLE, P. G. and SNELL, E. J. 1984. Random Walks and Electrical Networks. Math. Assoc. Amer., Washington, D.C.
  • GRIMMETT, G. R. KESTEN, H. and ZHANG, Y. 1993. Random walk on the infinite cluster of the percolation model. Probab. Theory Related Fields 96 33 44.
  • GRIMMETT, G. R. and MARSTRAND, J. M. 1990. The supercritical phase of percolation is well behaved. Proc. Roy. Soc. London Ser. A 430 439 457.
  • HAGGSTROM, O. MOSSEL, E. 1998. Nearest-neighbor walks with low predictability profile and ¨ ¨ percolation in 2 dimensions. Ann. Probab. 26 1212 1231.
  • HIEMER, P. 1998. Dynamical renormalisation in oriented percolation. Preprint.
  • HOFFMAN, C. 1998. Unpredictable nearest neighbor processes. Preprint.
  • KESTEN, H. and STIGUM, B. P. 1966. Additional limit theorems for indecomposable multidimensional Galton Watson processes. Ann. Math. Statist. 37 1463 1481.
  • KESTEN, H. and ZHANG, Y. 1990. The probability of a large finite cluster in supercritical Bernoulli percolation. Ann. Probab. 18 537 555.
  • LAWLER, G. 1991. Intersections of Random Walks. Birkhauser, Boston. ¨
  • LEVIN, D. and PERES, Y. 1998. Energy and cutsets in infinite percolation clusters. In Proceedings of the Cortona Workshop on Random Walks and Discrete Potential TheoryM. Picardello and W. Woess eds.. To appear. Z.
  • LIGGETT, T. M. SCHONMANN, R. H. and STACEY, A. M. 1996. Domination by product measures. Ann. Probab. 24 1711 1726.
  • LYONS, R. 1990. Random walks and percolation on trees. Ann. Probab. 18 931 958.
  • LYONS, R. 1995. Random walks and the growth of groups. C. R. Acad. Sci. Paris 320 1361 1366.
  • MOORE, T. and SNELL, J. L. 1979. A branching process showing a phase transition. J. Appl. Probab. 16 252 260.
  • PEMANTLE, R. and PERES, Y. 1996. On which graphs are all random walks in random environments transient? In Random Discrete Structures D. Aldous and R. Pemantle, eds. 207 211. Springer, New York.
  • PISZTORA, A. 1996. Surface order large deviation for Ising, Potts and percolation models. Probab. Theory Related Fields 104 427 466.
  • SOARDI, P. M. 1994. Potential Theory on Infinite Networks. Springer, Berlin.
  • MADISON, WISCONSIN 53706 E-MAIL: pemantle@math.wisc.edu
  • BERKELEY, CALIFORNIA