Electronic Journal of Probability

Frogs on trees?

Jonathan Hermon

Abstract

We study a system of simple random walks on $\mathcal{T} _{d,n}=({\cal V}_{d,n},{\cal E}_{d,n})$, the $d$-ary tree of depth $n$, known as the frog model. Initially there are Pois($\lambda$) particles at each site, independently, with one additional particle planted at some vertex $\mathbf{o}$. Initially all particles are inactive, except for the ones which are placed at $\mathbf{o}$. Active particles perform independent simple random walk on the tree of length $t \in{\mathbb N} \cup \{\infty \}$, referred to as the particles’ lifetime. When an active particle hits an inactive particle, the latter becomes active. The model is often interpreted as a model for a spread of an epidemic. As such, it is natural to investigate whether the entire population is eventually infected, and if so, how quickly does this happen. Let $\mathcal{R} _t$ be the set of vertices which are visited by the process (with lifetime $t$). The susceptibility ${\mathcal S}({\mathcal T}_{d,n}):=\inf \{t:\mathcal{R} _t={\cal V}_{d,n} \}$ is the minimal lifetime required for the process to visit all sites. The cover time $\mathrm{CT} ({\mathcal T}_{d,n})$ is the first time by which every vertex was visited at least once, when we take $t=\infty$. We show that there exist absolute constants $c,C>0$ such that for all $d \ge 2$ and all $\lambda = {\lambda }_n >0$ which does not diverge nor vanish too rapidly as a function of $n$, with high probability $c \le \lambda{\mathcal S} ({\mathcal T}_{d,n}) /[n\log (n / {\lambda } )] \le C$ and $\mathrm{CT} ({\mathcal T}_{d,n})\le 3^{4\sqrt{ \log |{\cal V}_{d,n}| } }$.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 17, 40 pp.

Dates
Received: 2 October 2016
Accepted: 25 January 2018
First available in Project Euclid: 23 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1519354946

Digital Object Identifier
doi:10.1214/18-EJP144

Mathematical Reviews number (MathSciNet)
MR3771754

Zentralblatt MATH identifier
1390.60351

Citation

Hermon, Jonathan. Frogs on trees?. Electron. J. Probab. 23 (2018), paper no. 17, 40 pp. doi:10.1214/18-EJP144. https://projecteuclid.org/euclid.ejp/1519354946