The Heavy-tailed Frog Model

Abstract

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 \[ Q(x,y)\asymp |x-y|^{-(d+\alpha)}, \qquad \alpha>0. \] Starting from an i.i.d. Poisson$(\lambda)$ 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 $\alpha\in(0,d)$, the time required to activate all vertices within distance $L$ of the origin is, with high probability, \[ (\log L)^{\Delta+o(1)}, \qquad \Delta^{-1}:=\log_2\left(\frac{2d}{d+\alpha}\right), \] 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 Ram\'irez and Sidoravicius (2004). When $\alpha>d$, we recover this classical linear behavior by proving matching linear upper and lower bounds; at $\alpha=d$, 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$(\lambda L^d)$ cloud of independent stationary random walkers.