The Annals of Probability

The number of open paths in oriented percolation

Olivier Garet, Jean-Baptiste Gouéré, and Régine Marchand

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We study the number $N_{n}$ of open paths of length $n$ in supercritical oriented percolation on $\mathbb{Z}^{d}\times\mathbb{N}$, with $d\ge1$, and we prove the existence of the connective constant for the supercritical oriented percolation cluster: on the percolation event $\{\inf N_{n}>0\}$, $N_{n}^{1/n}$ almost surely converges to a positive deterministic constant.

The proof relies on the introduction of adapted sequences of regenerating times, on subadditive arguments and on the properties of the coupled zone in supercritical oriented percolation. This global convergence result can be deepened to give directional limits and can be extended to more general random linear recursion equations known as linear stochastic evolutions.

Article information

Ann. Probab., Volume 45, Number 6A (2017), 4071-4100.

Received: November 2015
Revised: September 2016
First available in Project Euclid: 27 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B43: Percolation [See also 60K35]

Subadditive ergodic theorem oriented percolation


Garet, Olivier; Gouéré, Jean-Baptiste; Marchand, Régine. The number of open paths in oriented percolation. Ann. Probab. 45 (2017), no. 6A, 4071--4100. doi:10.1214/16-AOP1158.

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  • [1] Comets, F., Fukushima, R., Nakajima, S. and Yoshida, N. (2015). Limiting results for the free energy of directed polymers in random environment with unbounded jumps. J. Stat. Phys. 161 577–597.
  • [2] Comets, F., Popov, S. and Vachkovskaia, M. (2008). The number of open paths in an oriented $\rho$-percolation model. J. Stat. Phys. 131 357–379.
  • [3] Comets, F., Shiga, T. and Yoshida, N. (2004). Probabilistic analysis of directed polymers in a random environment: A review. In Stochastic Analysis on Large Scale Interacting Systems. Adv. Stud. Pure Math. 39 115–142. Math. Soc. Japan, Tokyo.
  • [4] Darling, R. W. R. (1991). The Lyapunov exponent for products of infinite-dimensional random matrices. In Lyapunov Exponents (Oberwolfach, 1990). Lecture Notes in Math. 1486 206–215. Springer, Berlin.
  • [5] Durrett, R. (1991). The contact process, 1974–1989. In Mathematics of Random Media (Blacksburg, VA, 1989). Lectures in Applied Mathematics 27 1–18. Amer. Math. Soc., Providence, RI.
  • [6] Fukushima, R. and Yoshida, N. (2012). On exponential growth for a certain class of linear systems. ALEA Lat. Am. J. Probab. Math. Stat. 9 323–336.
  • [7] Garet, O. and Marchand, R. (2012). Asymptotic shape for the contact process in random environment. Ann. Appl. Probab. 22 1362–1410.
  • [8] Garet, O. and Marchand, R. (2014). Large deviations for the contact process in random environment. Ann. Probab. 42 1438–1479.
  • [9] Kesten, H. and Sidoravicius, V. (2010). A problem in last-passage percolation. Braz. J. Probab. Stat. 24 300–320.
  • [10] Lacoin, H. (2012). Existence of an intermediate phase for oriented percolation. Electron. J. Probab. 17 no. 41, 17.
  • [11] Lacoin, H. (2014). Non-coincidence of quenched and annealed connective constants on the supercritical planar percolation cluster. Probab. Theory Related Fields 159 777–808.
  • [12] Yoshida, N. (2008). Phase transitions for the growth rate of linear stochastic evolutions. J. Stat. Phys. 133 1033–1058.