Electronic Journal of Probability

Frogs on trees?

Jonathan Hermon

Full-text: Open access

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

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 05C81: Random walks on graphs

Keywords
frog model epidemic spread rumor spread simple random walks cover times susceptibility trees

Rights
Creative Commons Attribution 4.0 International License.

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


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