The Annals of Applied Probability

The Williams–Bjerknes model on regular trees

Oren Louidor, Ran Tessler, and Alexander Vandenberg-Rodes

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We consider the Williams–Bjerknes model, also known as the biased voter model on the $d$-regular tree $\mathbb{T} ^{d}$, where $d\geq3$. Starting from an initial configuration of “healthy” and “infected” vertices, infected vertices infect their neighbors at Poisson rate $\lambda\geq1$, while healthy vertices heal their neighbors at Poisson rate $1$. All vertices act independently. It is well known that starting from a configuration with a positive but finite number of infected vertices, infected vertices will continue to exist at all time with positive probability if and only if $\lambda>1$. We show that there exists a threshold $\lambda_{c}\in(1,\infty)$ such that if $\lambda>\lambda_{c}$ then in the above setting with positive probability, all vertices will become eventually infected forever, while if $\lambda<\lambda_{c}$, all vertices will become eventually healthy with probability $1$. In particular, this yields a complete convergence theorem for the model and its dual, a certain branching coalescing random walk on $\mathbb{T} ^{d}$—above $\lambda_{c}$. We also treat the case of initial configurations chosen according to a distribution which is invariant or ergodic with respect to the group of automorphisms of $\mathbb{T} ^{d}$.

Article information

Ann. Appl. Probab., Volume 24, Number 5 (2014), 1889-1917.

First available in Project Euclid: 26 June 2014

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

Williams–Bjerknes biased voter local survival fixation


Louidor, Oren; Tessler, Ran; Vandenberg-Rodes, Alexander. The Williams–Bjerknes model on regular trees. Ann. Appl. Probab. 24 (2014), no. 5, 1889--1917. doi:10.1214/13-AAP966.

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  • [1] Biggins, J. D. (1995). The growth and spread of the general branching random walk. Ann. Appl. Probab. 5 1008–1024.
  • [2] Bramson, M. and Griffeath, D. (1980). On the Williams–Bjerknes tumour growth model. II. Math. Proc. Cambridge Philos. Soc. 88 339–357.
  • [3] Bramson, M. and Griffeath, D. (1981). On the Williams–Bjerknes tumour growth model. I. Ann. Probab. 9 173–185.
  • [4] Burkholder, D. L., Daley, D., Kesten, H., Ney, P., Spitzer, F., Hammersley, J. M. and Kingman, J. F. C. (1973). Discussion on “Subadditive ergodic theory” by J. F. C. Kingman. Ann. Probab. 1 900–909.
  • [5] Durrett, R. and Schinazi, R. (1995). Intermediate phase for the contact process on a tree. Ann. Probab. 23 668–673.
  • [6] Griffeath, D. (1979). Additive and Cancellative Interacting Particle Systems. Lecture Notes in Math. 724. Springer, Berlin.
  • [7] Harris, T. E. (1976). On a class of set-valued Markov processes. Ann. Probab. 4 175–194.
  • [8] Harris, T. E. (1978). Additive set-valued Markov processes and graphical methods. Ann. Probab. 6 355–378.
  • [9] Lalley, S. P. and Sellke, T. (1998). Limit set of a weakly supercritical contact process on a homogeneous tree. Ann. Probab. 26 644–657.
  • [10] Liggett, T. M. (1996). Multiple transition points for the contact process on the binary tree. Ann. Probab. 24 1675–1710.
  • [11] Liggett, T. M. (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Grundlehren der Mathematischen Wissenschaften 324. Springer, Berlin.
  • [12] Liggett, T. M. (2004). Interacting Particle Systems. Springer, Berlin.
  • [13] Lyons, R. and Peres, Y. (2014). Probability on Trees and Networks. Cambridge Univ. Press. To appear.
  • [14] Madras, N. and Schinazi, R. (1992). Branching random walks on trees. Stochastic Process. Appl. 42 255–267.
  • [15] Morgan, B. J. T. (1979). Four approaches to solving the linear birth-and-death (and similar) processes. Internat. J. Math. Ed. Sci. Tech. 10 51–64.
  • [16] Pemantle, R. (1992). The contact process on trees. Ann. Probab. 20 2089–2116.
  • [17] Richardson, D. (1973). Random growth in a tessellation. Math. Proc. Cambridge Philos. Soc. 74 515–528.
  • [18] Salzano, M. and Schonmann, R. H. (1998). A new proof that for the contact process on homogeneous trees local survival implies complete convergence. Ann. Probab. 26 1251–1258.
  • [19] Schwartz, D. L. (1977). Applications of duality to a class of Markov processes. Ann. Probab. 5 522–532.
  • [20] Stacey, A. M. (1996). The existence of an intermediate phase for the contact process on trees. Ann. Probab. 24 1711–1726.
  • [21] Sudbury, A. and Lloyd, P. (1997). Quantum operators in classical probability theory. IV. Quasi-duality and thinnings of interacting particle systems. Ann. Probab. 25 96–114.
  • [22] Williams, T. and Bjerknes, R. (1971). A Stochastic Model for the Spread of an Abnormal Clone Through the Basal Layer of the Epithelium. Symp. Tobacco Research Council, London.
  • [23] Zhang, Y. (1996). The complete convergence theorem of the contact process on trees. Ann. Probab. 24 1408–1443.