Rate of divergence of time constant for frog model with vanishing initial density

The frog model with a Bernoulli initial configuration is an interacting particle system on the $d$-dimensional lattice ($d \geq 2$) with two types of particles: active and sleeping. Active particles perform independent simple random walks. In contrast, although the sleeping particles do not move at first, they become active and start moving once touched by the active particles. Initially, only the origin has a single active particle, and the other sites have sleeping particles according to a Bernoulli distribution. After the original active particle starts moving, further active particles are gradually generated under the above rule and propagate across the lattice. The time required for the propagation of active frogs is expected to increase as the parameter of the Bernoulli distribution decreases, since fewer frogs are available. The aim of this paper is to investigate this increase in the vanishing density limit. In particular, we observe that it diverges and the rate of divergence differs significantly between $d=2$ and $d \geq 3$.

💡 Research Summary

The paper investigates how the time constant of the frog model on the $d$‑dimensional integer lattice behaves when the initial density of sleeping frogs, governed by a Bernoulli$(r)$ distribution, tends to zero. In the frog model, a single active frog starts at the origin and performs an independent simple random walk; whenever it visits a site containing sleeping frogs, those frogs become active and start their own independent walks. The first‑passage time $T(0,y)$ from the origin to a site $y$ is defined as the minimal time needed for any active frog to reach $y$. Prior work has shown that $T(0,nx)/n$ converges almost surely to a deterministic norm $\mu_r(x)$, called the time constant, which determines the asymptotic shape of the visited set.

The authors’ main goal is to determine the rate at which $\mu_r(x)$ diverges as $r\downarrow0$. They introduce a dimension‑dependent scaling function
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