Active phase for activated random walks on $\mathbb{Z}^d$, $ d \geq 3$, with density less than one and arbitrary sleeping rate

Lorenzo Taggi

Active phase for activated random walks on $\mathbb{Z}^d$, $ d \geq 3$, with density less than one and arbitrary sleeping rate

Probability 2021-05-31 v4

Authors: Lorenzo Taggi

Abstract

It has been conjectured that the critical density of the Activated Random Walk model is strictly less than one for any value of the sleeping rate. We prove this conjecture on $\mathbb{Z}^d$ when $d \geq 3$ and, more generally, on graphs where the random walk is transient. Moreover, we establish the occurrence of a phase transition on non-amenable graphs, extending previous results which require that the graph is amenable or a regular tree.

Keywords

activated random walks critical density sleeping rate phase transition transient random walk non-amenable graphs

Cite

@article{arxiv.1712.05292,
  title  = {Active phase for activated random walks on $\mathbb{Z}^d$, $ d \geq 3$, with density less than one and arbitrary sleeping rate},
  author = {Lorenzo Taggi},
  journal= {arXiv preprint arXiv:1712.05292},
  year   = {2021}
}

Comments

15 pages. arXiv admin note: text overlap with arXiv:1512.02397, Annales de l'Institut Henri Poincare (B) Probability and Statistics (2018)

Active phase for activated random walks on $\mathbb{Z}^d$, $ d \geq 3$, with density less than one and arbitrary sleeping rate

Abstract

Keywords

Cite

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