×

zbMATH — the first resource for mathematics

Growth profile and invariant measures for the weakly supercritical contact process on a homogeneous tree. (English) Zbl 0954.60090
Isotropic contact process on an infinite homogeneous tree of degree at least three is investigated. It is a continuous Markov process \(A_t\), where \(A_t\) are finite subsets of infected sites of the tree. The initial value is the root \(e\) of the tree. The switch from \(A_t\) to its complement happens at rate 1 and the switch from the complement of \(A_t\) to \(A_t\) at rate \(\lambda\) times the number of infected neighbours of \(t\). It is known that there are three phases under these circumstances, the extinction, weak survival, and strong survival. The paper examines the case, where for some interval \((\lambda_1,\lambda_2]\) of \(\lambda\)’s the weakly supercritical contact process appears (i.e. the weak survival occurs). It means more exactly that \(|A_t|\to\infty\) with positive probability and \(A_t\cap B\to\emptyset\) for finite subsets \(B\) of the tree almost surely. Two important quantities are the exponential rate \(\beta\) of decay of hitting probability function at \(\infty\), i.e. \(\beta=\lim_{n=d(x,e)\to\infty} (P\{(\exists t) x\in A_t\})^{{1}/{n}}\) and the growth profile \(V(s)=\log\lim_{n=d(x,e)\to\infty} (P\{x\in A_{ns}\})^{1/n}\). It is shown that \(\beta\) is strictly increasing for \(\lambda \in (\lambda_1,\lambda_2]\), that \(V(s)\) is concave, continuous, at most \(\log\beta\), and the asymptotic behaviour of \(V(s)\) at \(0_+\) and at \(\infty\) is described. Liggett’s function \(\varphi\) that enables to recognize the existence of a spherically invariant measure for the contact process that satisfies certain exponential decay law is described in terms of \(V(s)\). Using the mentioned results and the result of S. P. Lalley and T. Sellke [ibid. 26, No. 2, 644-657 (1998; Zbl 0937.60093)] that \(\beta\leq {1}/{\sqrt{d}}\) for the weakly supercritical process, the conjecture that a spherically invariant measure with the exponential decay exists is verified if \(\lambda\in (\lambda_1,\lambda_2)\).
Reviewer: P.Holicky (Praha)

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] BEZUIDENHOUT, C. and GRIMMETT, G. 1991. Exponential decay for subcritical contact and percolation processes. Ann. Probab. 19 984 1009. · Zbl 0743.60107 · doi:10.1214/aop/1176990332
[2] BOLLOBAS, B. 1979. Graph Theory. Springer, New York. · Zbl 0411.05032
[3] DURRETT, R. and SCHINAZI, R. 1995. Intermediate phase for the contact process on a tree. Ann. Probab. 23 668 673. · Zbl 0830.60093 · doi:10.1214/aop/1176988283
[4] GRIMMETT, G. 1989. Percolation. Springer, New York. · Zbl 0691.60089
[5] HARRIS, T. E. 1978. Additive set-valued Markov processes and percolation methods. Ann. Probab. 6 355 378. · Zbl 0378.60106 · doi:10.1214/aop/1176995523
[6] LALLEY, S. and SELLKE, T. 1995. Hyperbolic branching Brownian motion. Probab. Theory Related Fields 108 171 192. · Zbl 0883.60092 · doi:10.1007/s004400050106
[7] LALLEY, S. and SELLKE, T. 1998. Limit set of a weakly supercritical contact process on a homogeneous tree. Ann. Probab. 26 644 657. · Zbl 0937.60093 · doi:10.1214/aop/1022855646
[8] LIGGETT, T. 1985. Interacting Particle Systems. Springer, New York. · Zbl 0559.60078
[9] LIGGETT, T. 1996. Branching random walks and contact processes on homogeneous trees. Probab. Theory Related Fields 106 495 519. · Zbl 0867.60092 · doi:10.1007/s004400050073
[10] LIGGETT, T. 1996. Multiple transition points for the contact process on the binary tree. Ann. Probab. 24 1675 1710. · Zbl 0871.60087 · doi:10.1214/aop/1041903202
[11] LIGGETT, T. 1996. Stochastic models of interacting systems. Ann. Probab. 25 1 29.
[12] LOVASZ, L. 1993. Combinatorial Problems and Exercises. North-Holland, Amsterdam.
[13] MENSHIKOV, M. 1986. Coincidence of critical points in percolation problems. Soviet Math. Dokl. 33 856 859. · Zbl 0615.60096
[14] PEMANTLE, R. 1992. The contact process on trees. Ann. Probab. 20 2089 2116. · Zbl 0762.60098 · doi:10.1214/aop/1176989541
[15] SCHONMANN, R. 1998. The triangle condition for contact processes on homogeneous trees. J. Statist. Phys. 90 1429 1440. · Zbl 0922.60087 · doi:10.1023/A:1023247932037
[16] STACEY, A. M. 1996. Existence of an intermediate phase for the contact process on trees. Ann. Probab. 24 1711 1726. · Zbl 0878.60061 · doi:10.1214/aop/1041903203
[17] ZHANG, Y. 1996. The complete convergence theorem of the contact process on trees. Ann. Probab. 24 1408 1443. · Zbl 0876.60092 · doi:10.1214/aop/1065725187
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.