The Annals of Probability

Critical exponents for a reversible nearest particle system on the binary tree

Amber L. Puha

Abstract

The uniform model is a reversible interactingparticle system that evolves on the homogeneous tree. Occupied sites become vacant at rate one provided the number of occupied neighbors does not exceed one.Vacant sites become occupied at rate $\beta$ times the number of occupied neighbors. On the binary tree, it has been shown that the survival threshold $\beta_c$ is 1/4. In particular, for $\beta \leq 1/4$, the expected extinction time is finite.Otherwise, the uniform model survives locally. We show that the survival probability decays faster than a quadratic near $\beta_c$. This contrasts with the behavior of the survival probability for the contact process on homogeneous trees, which decays linearly.We also provide a lower bound that implies that the rate of decay is slower than a cubic. Tools associated with reversibility, for example, the Dirichlet principle and Thompson's principle, are used to prove this result.

Article information

Source
Ann. Probab., Volume 28, Number 1 (2000), 395-415.

Dates
First available in Project Euclid: 18 April 2002

https://projecteuclid.org/euclid.aop/1019160124

Digital Object Identifier
doi:10.1214/aop/1019160124

Mathematical Reviews number (MathSciNet)
MR1756010

Zentralblatt MATH identifier
1044.60097

Citation

Puha, Amber L. Critical exponents for a reversible nearest particle system on the binary tree. Ann. Probab. 28 (2000), no. 1, 395--415. doi:10.1214/aop/1019160124. https://projecteuclid.org/euclid.aop/1019160124

References

• [1] Barsky, D. and Wu, C. (1998). Critical exponents for the contact process under the triangle condition. J. Statist. Phys. 91 95-124.
• [2] Chen, D. (1988). On the survival probability of generalized nearest particle systems. Stochastic Process. Appl. 30 209-223.
• [3] Chen, D. (1994). Finite nearest particle systems on a tree. Acta Math. Sci. 14 348-353.
• [4] Doyle, P. and Snell, J. (1984). Random Walk and Electric Networks. Math. Assoc. America, Washington, DC.
• [5] Griffeath, D. and Liggett, T. (1982). Critical phenomenon for Spitzer's reversible nearestparticle system. Ann. Probab. 10 881-895.
• [6] Liggett, T. (1985). Interacting Particle Systems. Springer, New York.
• [7] Liggett, T. (1985). Reversible growth models on symmetric sets. Proceedings of the Taniguchi Symposium 275-301.
• [8] Liggett, T. (1987). Applications of the Dirichlet principle to finite reversible nearest particle systems. Probab. Theory Related Fields 74 505-528.
• [9] Liggett, T. (1987). Reversible growth models on d: some examples. In Percolation Theory and Ergodic Theory of Infinite Particle Systems 213-227. Springer, New York.
• [10] Liggett, T. (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, New York.
• [11] Puha, A. (1998). A reversible interactingparticle system on the homogeneous tree. Ph.D. Dissertation, Dept. of Math., Univ. California, Los Angeles.
• [12] Puha, A. (1999). A reversible nearest particle system on the homogeneous tree. J. Theoret. Probab. 12 217-254.
• [13] Salzano, M. and Schonmann, R. (1998). A new proof that for the contact process on homogeneous trees local survival implies complete convergence. Ann. Probab. 26 1251-1258.
• [14] Schonmann, R. (1998). The triangle condition for contact processes on homogeneous trees. J. Statist. Phys. 90 1429-1440.
• [15] Spitzer, F. (1977). Stochastic time evolution of one-dimensional infinite-particle systems. Bull. Amer. Math. Soc. 83 880-890.
• [16] Tretyakov, A. and Konno, N. (1999). Numerical estimation on critical exponent for uniform model on binary tree. In Proceedings of the International Workshop on Soft Computing in Industry.
• [17] Wu, C. (1995). The contact process on a tree: behavior near the first transition. Stochastic Process. Appl. 57 99-112.
• [18] Liggett, T. (2000). Monotonicity of condition distributions and growth models on trees. Ann. Probab. To appear.