论文标题
限制双曲线组的分支随机步行
Limit set of branching random walks on hyperbolic groups
论文作者
论文摘要
令$γ$是一个非元素双曲线组，单词度量$ d $和$ \partialγ$其双曲线边界配备了视觉度量$ d_a $，用于某些参数$ a> 1 $。修复$γ$上的superexponential对称概率$ $ $ $ $ $，其支持会产生$γ$作为半群，并用$ρ$表示$§y $ y $ y $ y $γ$的频谱半径，带有step Distribution $μ$。令$ \ {1，\，2，\，3，\，\ ldots \} $，$ \ {1，\，\，\ ldots \} $的概率为$ \ sum = \ sum \ limits_ {k = 1}^\ inftykν（k）<\ infty $。 令$ \ mathrm {brw}（γ，\，\，ν，\，μ）$为$γ$上的分支随机步行，带有后代分布$ν$和基本运动$ y $ y $ y $ y $ y $ y $和$ h（λ）$的量$ \ \ mathrm {brw}（γ，\，\，c）的体积增长率。我们证明了$λ\在[1，\，ρ^{ - 1}）$中，即极限设置的hausdorff尺寸$λ$，这是$（\partialγ，\，d_a）$的随机子集，由$ \ mathrm {brw}（brw}（γ）（γ）的所有积累点组成， $ \ log_a h（λ）$。此外，我们证明$ h（λ）$几乎是确定性的，严格的，严格增加$λ\在[1，\，ρ^{ - 1} $中的$λ\，都受体积增长率的平方根为$γ$的范围，并且具有关键的指数$ 1/2 $ $ 1/2 $ $ 1/2 $ $ρ^{ - 1} $ρ^{ - 1} $ extement。 \ [ h（ρ^{ - 1}） - h（λ）\ sim c \ sqrt {ρ^{ - 1} -c \]对于某些正恒定$ c $。我们推测关键情况下的$λ$的hausdorff尺寸几乎可以肯定。这已在有限的许多有限组的自由组或免费的产品（通过合并）上确认，该组配备了标准生成集定义的标准$ d $。
Let $Γ$ be a nonelementary hyperbolic group with a word metric $d$ and $\partialΓ$ its hyperbolic boundary equipped with a visual metric $d_a$ for some parameter $a>1$. Fix a superexponential symmetric probability $μ$ on $Γ$ whose support generates $Γ$ as a semigroup, and denote by $ρ$ the spectral radius of the random walk $Y$ on $Γ$ with step distribution $μ$. Let $ν$ be a probability on $\{1,\, 2, \, 3, \, \ldots\}$ with mean $λ=\sum\limits_{k=1}^\infty kν(k)<\infty$. Let $\mathrm{BRW}(Γ, \, ν, \, μ)$ be the branching random walk on $Γ$ with offspring distribution $ν$ and base motion $Y$ and $H(λ)$ the volume growth rate for the trace of $\mathrm{BRW}(Γ, \, ν, \, μ)$. We prove for $λ\in [1, \, ρ^{-1})$ that the Hausdorff dimension of the limit set $Λ$, which is the random subset of $(\partial Γ, \, d_a)$ consisting of all accumulation points of the trace of $\mathrm{BRW}(Γ, \, ν, \, μ)$, is given by $\log_a H(λ)$. Furthermore, we prove that $H(λ)$ is almost surely a deterministic, strictly increasing and continuous function of $λ\in [1, \, ρ^{-1}]$, is bounded by the square root of the volume growth rate of $Γ$, and has critical exponent $1/2$ at $ρ^{-1}$ in the sense that \[ H(ρ^{-1}) - H(λ) \sim C \sqrt{ρ^{-1} - λ} \quad \text{as } λ\uparrow ρ^{-1} \] for some positive constant $C$. We conjecture that the Hausdorff dimension of $Λ$ in the critical case $λ=ρ^{-1}$ is $\log_aH(ρ^{-1})$ almost surely. This has been confirmed on free groups or the free product (by amalgamation) of finitely many finite groups equipped with the word metric $d$ defined by the standard generating set.
