Anisotropic branching random walks on homogeneous trees

• 成果类型: Article
• 署名作者: Hueter, I; Lalley, SP
• 署名单位: State University System of Florida; University of Florida; University of Chicago
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/PL00008723
• 发表日期: 2000
• 页码: 57-88
• 关键词: contact process

摘要

Symmetric branching random walk on a homogeneous tree exhibits a weak survival phase: For parameter values in a certain interval, the population survives forever with positive probability, but, with probability one, eventually vacates every finite subset of the tree. In this phase, particle trails must converge to the geometric boundary Omega of the tree. The random subset Lambda of the boundary consisting of all ends of the tree in which the population 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. Moreover, there is strict inequality at the phase separation point between weak and strong survival except when the branching random walk is isotropic. It is further shown that in all cases there is a distinguished probability measure mu supported by Omega such that the Hausdorff dimension of Lambda boolean AND Omega(mu) , where Omega(mu) is the set of mu-generic points of Omega, converges to one half the Hausdorff dimension of Omega(mu) at the phase separation point. Exact formulas are obtained for the Hausdorff dimensions of Lambda and Lambda boolean AND Omega(mu) , and it is shown that the log Hausdorff dimension of Lambda has critical exponent 1/2 at the phase separation point.