Absorbing-state phase transition and activated random walks with unbounded capacities

Leandro Chiarini; Alexandre Stauffer

doi:10.30757/ALEA.V19-46

Absorbing-state phase transition and activated random walks with unbounded capacities

Leandro Chiarini, Alexandre Stauffer

University of Utrecht

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Abstract

In this article, we study the existence of an absorbing-state phase transition of an Abelian process that generalises the Activated Random Walk (ARW). Given a vertex transitive G = (V;E), we associate to each site x 2 V a capacity wx ≥ 0, which describes how many inactive particles x can hold, where fwxgx2V is a collection of i.i.d random variables. When G is an amenable graph, we prove that if E[wx] < 1, the model goes through an absorbing-state phase transition and if E[wx] = 1, the model fixates for all λ > 0. Moreover, in the former case, we provide bounds for the critical density that match the ones available in the classical Activated Random Walk.

Original language	English
Pages (from-to)	1123-1131
Number of pages	9
Journal	ALEA Latin American Journal of Probability and Mathematical Statistics
Volume	19
Issue number	2
Early online date	6 Aug 2022
DOIs	https://doi.org/10.30757/ALEA.V19-46
Publication status	Published - 31 Dec 2022

Bibliographical note

Funding Information:
Received by the editors August 13th, 2021; accepted June 13th, 2022. 2010 Mathematics Subject Classification. 82C22, 60K35, 60K37. Key words and phrases. Activated Random Walks, Random Environments, absorbing-state phase transition. The authors would like to thank the anonymous referee for many their suggestions to improve the exposition of this article. A.S. is supported by EPSRC Fellowship EP/N004566/1.

Keywords

Absorbing-state phase transition
Activated random walks
Random environments

ASJC Scopus subject areas

Statistics and Probability

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10.30757/ALEA.V19-46

2108.03038v1Submitted manuscript, 476 KB
arw-in-re (1).pdfAccepted author manuscript, 539 KB

Cite this

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title = "Absorbing-state phase transition and activated random walks with unbounded capacities",

abstract = "In this article, we study the existence of an absorbing-state phase transition of an Abelian process that generalises the Activated Random Walk (ARW). Given a vertex transitive G = (V;E), we associate to each site x 2 V a capacity wx ≥ 0, which describes how many inactive particles x can hold, where fwxgx2V is a collection of i.i.d random variables. When G is an amenable graph, we prove that if E[wx] < 1, the model goes through an absorbing-state phase transition and if E[wx] = 1, the model fixates for all λ > 0. Moreover, in the former case, we provide bounds for the critical density that match the ones available in the classical Activated Random Walk.",

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note = "Funding Information: Received by the editors August 13th, 2021; accepted June 13th, 2022. 2010 Mathematics Subject Classification. 82C22, 60K35, 60K37. Key words and phrases. Activated Random Walks, Random Environments, absorbing-state phase transition. The authors would like to thank the anonymous referee for many their suggestions to improve the exposition of this article. A.S. is supported by EPSRC Fellowship EP/N004566/1. ",

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AU - Stauffer, Alexandre

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N2 - In this article, we study the existence of an absorbing-state phase transition of an Abelian process that generalises the Activated Random Walk (ARW). Given a vertex transitive G = (V;E), we associate to each site x 2 V a capacity wx ≥ 0, which describes how many inactive particles x can hold, where fwxgx2V is a collection of i.i.d random variables. When G is an amenable graph, we prove that if E[wx] < 1, the model goes through an absorbing-state phase transition and if E[wx] = 1, the model fixates for all λ > 0. Moreover, in the former case, we provide bounds for the critical density that match the ones available in the classical Activated Random Walk.

AB - In this article, we study the existence of an absorbing-state phase transition of an Abelian process that generalises the Activated Random Walk (ARW). Given a vertex transitive G = (V;E), we associate to each site x 2 V a capacity wx ≥ 0, which describes how many inactive particles x can hold, where fwxgx2V is a collection of i.i.d random variables. When G is an amenable graph, we prove that if E[wx] < 1, the model goes through an absorbing-state phase transition and if E[wx] = 1, the model fixates for all λ > 0. Moreover, in the former case, we provide bounds for the critical density that match the ones available in the classical Activated Random Walk.

KW - Absorbing-state phase transition

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Absorbing-state phase transition and activated random walks with unbounded capacities

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

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Early Career Fellowship - Mathematical Analysis of Strongly Correlated Processes on Discrete Dynamic Structures

Cite this

Absorbing-state phase transition and activated random walks with unbounded capacities

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Projects

Early Career Fellowship - Mathematical Analysis of Strongly Correlated Processes on Discrete Dynamic Structures

Cite this