论文标题
在相对双曲的组上进行随机分支
Branching Random Walks on relatively hyperbolic groups
论文作者
论文摘要
令$γ$为具有有限生成集的非元素相对双曲线。考虑$γ$上的有限支持的可接受和对称的概率度量$ $ $ $ $ $ $ \ MATHBB {n} $的概率度量$ν$，带有平均$ r $。令$ \ mathrm {brw}（γ，γ，μ）$为$γ$上的分支随机步行，带有后代分布$ν$和带有步骤分配$μ$的随机步行给出的基本运动。众所周知，$ 1 <r \ leq r $带有$ r $的融合半径是随机步行的绿色功能，$ \ mathrm {brw}（γ，γ，γ，ν，μ）的人口永远存活，但最终使每一个有限的$γ$子集撤离。我们证明，在这种制度中，分支随机步行的痕迹的增长率等于基础随机步行的绿色功能的增长率$ω_γ（r）$。我们还证明，限制集合$λ（r）$的Hausdorff尺寸，这是Bowditch边界的随机子集，由$ \ MATHRM {BRW}（γ，γ，ν，μ）$组成的所有累积点组成，等于恒定的时间$ω______γ（r）$。
Let $Γ$ be a non-elementary relatively hyperbolic group with a finite generating set. Consider a finitely supported admissible and symmetric probability measure $μ$ on $Γ$ and a probability measure $ν$ on $\mathbb{N}$ with mean $r$. Let $\mathrm{BRW}(Γ,ν,μ)$ be the branching random walk on $Γ$ with offspring distribution $ν$ and base motion given by the random walk with step distribution $μ$. It is known that for $1 < r \leq R$ with $R$ the radius of convergence for the Green function of the random walk, the population of $\mathrm{BRW}(Γ,ν,μ)$ survives forever, but eventually vacates every finite subset of $Γ$. We prove that in this regime, the growth rate of the trace of the branching random walk is equal to the growth rate $ω_Γ(r)$ of the Green function of the underlying random walk. We also prove that the Hausdorff dimension of the limit set $Λ(r)$, which is the random subset of the Bowditch boundary consisting of all accumulation points of the trace of $\mathrm{BRW}(Γ,ν,μ)$, is equal to a constant times $ω_Γ(r)$.
