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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Abelian oil and water dynamics does not have an absorbing-state phase transition


Authors: Elisabetta Candellero, Alexandre Stauffer and Lorenzo Taggi
Journal: Trans. Amer. Math. Soc. 374 (2021), 2733-2752
MSC (2020): Primary 60K35, 82C22, 82C26
DOI: https://doi.org/10.1090/tran/8276
Published electronically: January 20, 2021
MathSciNet review: 4223031
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Abstract: The oil and water model is an interacting particle system with two types of particles and a dynamics that conserves the number of particles, which belongs to the so-called class of Abelian networks. Widely studied processes in this class are sandpiles models and activated random walks, which are known (at least for some choice of the underlying graph) to undergo an absorbing-state phase transition. This phase transition characterizes the existence of two regimes, depending on the particle density: a regime of fixation at low densities, where the dynamics converges towards an absorbing state and each particle jumps only finitely many times, and a regime of activity at large densities, where particles jump infinitely often and activity is sustained indefinitely. In this work we show that the oil and water model is substantially different than sandpiles models and activated random walks, in the sense that it does not undergo an absorbing-state phase transition and is in the regime of fixation at all densities. Our result works in great generality: for any graph that is vertex transitive and for a large class of initial configurations.


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Additional Information

Elisabetta Candellero
Affiliation: Dip. di Matematica e Fisica, Università Roma Tre, Largo S. Murialdo 1, 00146, Rome, Italy
MR Author ID: 968125
ORCID: 0000-0003-2424-8695
Email: ecandellero@mat.uniroma3.it

Alexandre Stauffer
Affiliation: Dip. di Matematica e Fisica, Università Roma Tre, Largo S. Murialdo 1, 00146, Rome, Italy; and Department of Mathematical Sciences, University of Bath, BA2 7AY Bath, United Kingdom
MR Author ID: 781199
Email: a.stauffer@bath.ac.uk

Lorenzo Taggi
Affiliation: Weierstrass Institute for applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany
Address at time of publication: Dipartimento di Matematica, Sapienza Università di Roma, Piazzale Aldo Moro 5, 00186, Roma, Italy
MR Author ID: 1106315
Email: lorenzo.taggi@uniroma1.it

Received by editor(s): January 30, 2019
Received by editor(s) in revised form: July 23, 2020
Published electronically: January 20, 2021
Additional Notes: The first author was partially supported by the project “Programma per Giovani Ricercatori Rita Levi Montalcini” awarded by the Italian Ministry of Education (MIUR). The first author also acknowledges partial support by “INdAM – GNAMPA Project 2019”.
The second and third authors acknowledge support from EPSRC Early Career Fellowship EP/N004566/1.
The third author acknowledges support from DFG German Research Foundation BE 5267/1.
Article copyright: © Copyright 2021 American Mathematical Society