arXiv 2007.132672020-07-27math.PRmath.DSmath.MG

双曲群上分支随机游动的极限集

Limit set of branching random walks on hyperbolic groups

Vladas Sidoravicius, Longmin Wang, Kainan Xiang

摘要：

设$Γ$是一个非初等双曲群，其字度量为$d$，$\partialΓ$是其双曲边界，对于某个参数$a>1$，其双曲边界具有可视度量$d_a$。在$Γ$上固定一个超指数对称概率$μ$，它的支撑生成$Γ$作为半群，用$ρ$表示$Γ$上的随机游动$Y$的谱半径，其阶跃分布为$μ$ .令$ν$为具有平均值$λ=\sum\limits_{k=1}^\infty kν(k)<\infty$的$\{1,\, 2, \, 3, \, \ldots\}$上的概率。令$\mathrm{BRW}(Γ, \, ν, \, μ)$为$Γ$上的分支随机游动，其后代分布为$ν$，基运动为$Y$，$H(λ)$为$\mathrm{BRW}(Γ, \, ν, \, μ)$迹的体积增长率。证明了$λ\in [1, \, ρ^{-1})$的极限集$Λ$的Hausdorff维数由$\log_a H(λ)$给出，$Λ$是由$\mathrm{BRW}(Γ, \, ν, \, μ)$的迹的所有聚点组成的$(\partial Γ, \, d_a)$的随机子集.进一步证明了$H(λ)$几乎必然是$λ\in [1, \, ρ^{-1}]$的一个确定性的、严格递增的连续函数，且以$Γ$的体积增长率的平方根为界，在$ρ^{-1}$处有临界指数$1/2$，在这个意义上，对于某个正的常数$C$，\[ H(ρ^{-1}) - H(λ) \sim C \sqrt{ρ^{-1} - λ} \quad \text{as } λ\uparrow ρ^{-1} \].我们猜想在临界情形$λ=ρ^{-1}$下$Λ$的Hausdorff维数几乎必然是$\log_aH(ρ^{-1})$.这已经在自由群或由标准生成集定义的具有字度量$d$的许多有限群的自由积（通过合并）上得到证实。

英文摘要：

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.

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