重尾青蛙模型
The Heavy-tailed Frog Model
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中文总结 AI 辅助
该研究针对重尾转移核下的青蛙模型，明确了不同尾指数区间的激活时间尺度，并揭示了有限寿命模型的覆盖寿命渐近特性，丰富了非经典随机游走驱动的传播过程理论。
中文摘要 AI 辅助
我们研究了d≥2维整数格点ℤᵈ和离散环面𝕋_L^d上的青蛙模型，其转移核对称、平移不变且具有重尾特性，满足Q(x,y)≍|x−y|^{-(d+α)}，其中α>0。初始时每个位点上有独立同分布的Poisson(λ)个休眠粒子，原点处有一个激活粒子。激活粒子执行独立的Q-随机游走，并激活其遇到的粒子。我们首先确定了激活远距离顶点的时间尺度。当α∈(0,d)时，随着L→∞，激活原点距离L范围内所有顶点所需的时间以高概率为(log L)^{Δ+o(1)}，其中Δ⁻¹:=log₂(2d/(d+α))。这种多对数级的传播速度与简单随机游走驱动的经典青蛙模型的线性传播形成鲜明对比，相关结果可参见Alves、Machado和Popov（2002）以及Ramírez和Sidoravicius（2004）的研究。当α>d时，我们通过证明匹配的线性上下界，重现了这种经典的线性行为；在α=d的情形下，我们证明了线性上界。最后，我们考虑了𝕋_L^d上的有限寿命模型，其中每个粒子在走ℓ步后被移除。我们证明，覆盖寿命（即环面被完全激活所需的最小ℓ值）渐近于Poisson(λLᵈ)个独立平稳随机游走者构成的粒子云的覆盖时间。
英文摘要
We study the frog model on $\mathbb Z^d$ and on the discrete tori $\mathbb T_L^d$, $d\ge 2$, with a symmetric, translation-invariant, and heavy-tailed transition kernel satisfying \[ Q(x,y)\asymp |x-y|^{-(d+α)}, \qquad α>0. \] Starting from an i.i.d. Poisson$(λ)$ number of sleeping particles per site and one active particle at the origin. Active particles perform independent $Q$-random walks and activate the particles they encounter. We first determine the timescale for activating distant vertices. When $α\in(0,d)$, the time required to activate all vertices within distance $L$ of the origin is, with high probability, \[ (\log L)^{Δ+o(1)}, \qquad Δ^{-1}:=\log_2\left(\frac{2d}{d+α}\right), \] as $L\to\infty$. This polylogarithmic spreading contrasts sharply with the linear spreading of the classical frog model driven by simple random walks; see Alves, Machado, and Popov (2002) and Ramírez and Sidoravicius (2004). When $α>d$, we recover this classical linear behavior by proving matching linear upper and lower bounds; at $α=d$, we prove a linear upper bound. Finally, we consider the finite-lifespan model on $\mathbb T_L^d$, in which each particle is removed after taking $\ell$ steps. We show that the cover lifespan, defined as the smallest $\ell$ for which the torus is entirely activated, is asymptotic to the cover time of a Poisson$(λL^d)$ cloud of independent stationary random walkers.
