2001/09/06 by Irene Hueter, Hueter, Irene
Mathematics · Physics and Astronomy · #28A80 (Secondary) #60K35 (Primary) 60J80 #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math-ph #math.MP #math.PR #msc:28A80 #msc:60J80 #msc:60K35
paper · pdf · doi:10.48550/arxiv.math/0109047
arxiv created 2001/09/06 · openalex publication_date 2001/09/06 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The existence of a weak survival region is established for the anisotropic symmetric contact process on a homogeneous tree T2d of degree 2d > 2: For parameter values in a certain connected region of positive Lebesgue measure, the population survives forever with positive probability but ultimately vacates every finite subset of the tree with probability one. In this phase, infection trails must converge to the geometric boundary Ωof the tree. The random subset Λof the boundary consisting of all ends of the tree in which the infection survives, called the limit set of the process, is shown to have Hausdorff dimension no larger than one half the Hausdorff dimension of the entire geometric boundary. In addition, there is strict inequality at the transition between weak and strong survival except when the contact process is isotropic. It is further shown that in all cases there is a distinguished probability measure μ, supported by Ω, such that the Hausdorff dimension of Λ∩ Ωμ, where Ωμ is the set of μ-generic points of Ω, converges to one half the Hausdorff dimension of Ωμ at the phase separation points. Exact formulae for the Hausdorff dimensions of Λand Λ∩ Ωμ are obtained. We also prove that the contact process at the transition between extinction and weak survival does not survive. The method developed shows that the contact process at the phase transition to strong survival survives weakly for d > 1.