We study the frog model on Cayley graphs of groups with polynomial growth rate D ≥ 3. The frog model is an interacting particle system in discrete time. We consider that the process begins with a particle at each vertex of the graph and only one of these particles is active when the process begins. Each activated particle performs a simple random walk in discrete time activating the inactive particles in the visited vertices. We prove that the activation time of particles grows at least linearly and we show that in the abelian case with any finite generator set the set of activated sites has a limiting shape.
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DOI : 10.1051/ps/2021007
Keywords: Shape theorem, frog model, Cayley graph, interacting particle system
@article{PS_2021__25_1_204_0,
author = {Coletti, Cristian F. and de Lima, Lucas R.},
title = {The asymptotic shape theorem for the frog model on finitely generated abelian groups},
journal = {ESAIM: Probability and Statistics},
pages = {204--219},
year = {2021},
publisher = {EDP-Sciences},
volume = {25},
doi = {10.1051/ps/2021007},
mrnumber = {4234132},
zbl = {1468.60115},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ps/2021007/}
}
TY - JOUR AU - Coletti, Cristian F. AU - de Lima, Lucas R. TI - The asymptotic shape theorem for the frog model on finitely generated abelian groups JO - ESAIM: Probability and Statistics PY - 2021 SP - 204 EP - 219 VL - 25 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/ps/2021007/ DO - 10.1051/ps/2021007 LA - en ID - PS_2021__25_1_204_0 ER -
%0 Journal Article %A Coletti, Cristian F. %A de Lima, Lucas R. %T The asymptotic shape theorem for the frog model on finitely generated abelian groups %J ESAIM: Probability and Statistics %D 2021 %P 204-219 %V 25 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/ps/2021007/ %R 10.1051/ps/2021007 %G en %F PS_2021__25_1_204_0
Coletti, Cristian F.; de Lima, Lucas R. The asymptotic shape theorem for the frog model on finitely generated abelian groups. ESAIM: Probability and Statistics, Tome 25 (2021), pp. 204-219. doi: 10.1051/ps/2021007
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Cité par Sources :
This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – Finance Code 001.
The first author was partially supported by grant #2017/10555-0, São Paulo Research Foundation (FAPESP).
