TY - JOUR

T1 - Abelian oil and water dynamics does not have an absorbing-state phase transition

AU - Candellero, Elisabetta

AU - Stauffer, Alexandre

AU - Taggi, Lorenzo

N1 - Funding Information:
Received by the editors January 30, 2019, and, in revised form, July 23, 2020. 2020 Mathematics Subject Classification. Primary 60K35, 82C22, 82C26. 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.
Publisher Copyright:
© 2021 American Mathematical Society

PY - 2021/5/1

Y1 - 2021/5/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85102118674&partnerID=8YFLogxK

U2 - 10.1090/tran/8276

DO - 10.1090/tran/8276

M3 - Article

VL - 374

SP - 2733

EP - 2752

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 4

ER -