【摘 要】
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Symmetric branching random walk on a Caylay graph of non-elementary hyperbolic group exhibits a weak survival phase: For growth parameter $lambda$ in the i
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Symmetric branching random walk on a Caylay graph of non-elementary hyperbolic group exhibits a weak survival phase: For growth parameter $\lambda$ in the interval $[1,\,R]$ where $R$ is the convergence radius of the underlying random walk,the population survives forever with positive probability,but with probability one,eventually vacates every finite subset of the graph.In this phase,particle trails must converge to a random subset $\Lambda$ in the geometric boundary of the graph.It is conjectured that the Hausdorff dimension $\Phi(\lambda)$ of the random set $\Lambda$ has the following behavior as $\lambda \uparrow R$: $\Phi(R)-\Phi(\lambda)\sim C \sqrt{R-\lambda}$ for some positive constant $C$.In this talk,we will review some progress of this conjecture on Fuchsian groups.
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