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July 2018 Critical density of activated random walks on transitive graphs
Alexandre Stauffer, Lorenzo Taggi
Ann. Probab. 46(4): 2190-2220 (July 2018). DOI: 10.1214/17-AOP1224

Abstract

We consider the activated random walk model on general vertex-transitive graphs. A central question in this model is whether the critical density $\mu_{c}$ for sustained activity is strictly between 0 and 1. It was known that $\mu_{c}>0$ on $\mathbb{Z}^{d}$, $d\geq1$, and that $\mu_{c}<1$ on $\mathbb{Z}$ for small enough sleeping rate. We show that $\mu_{c}\to0$ as $\lambda\to0$ in all vertex-transitive transient graphs, implying that $\mu_{c}<1$ for small enough sleeping rate. We also show that $\mu_{c}<1$ for any sleeping rate in any vertex-transitive graph in which simple random walk has positive speed. Furthermore, we prove that $\mu_{c}>0$ in any vertex-transitive amenable graph, and that $\mu_{c}\in(0,1)$ for any sleeping rate on regular trees.

Citation

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Alexandre Stauffer. Lorenzo Taggi. "Critical density of activated random walks on transitive graphs." Ann. Probab. 46 (4) 2190 - 2220, July 2018. https://doi.org/10.1214/17-AOP1224

Information

Received: 1 December 2015; Revised: 1 June 2017; Published: July 2018
First available in Project Euclid: 13 June 2018

zbMATH: 06919022
MathSciNet: MR3813989
Digital Object Identifier: 10.1214/17-AOP1224

Subjects:
Primary: 60K35 , 82C22
Secondary: 82C26

Keywords: absorbing states phase transitions , interacting particle systems , Random walks

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 4 • July 2018
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