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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 socalled 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 absorbingstate 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 absorbingstate 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.
Original language  English 

Pages (fromto)  27332752 
Number of pages  20 
Journal  Transactions of the American Mathematical Society 
Volume  374 
Issue number  4 
Early online date  20 Jan 2021 
DOIs  
Publication status  Published  1 May 2021 
Bibliographical note
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
ASJC Scopus subject areas
 General Mathematics
 Applied Mathematics
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 1 Finished

Early Career Fellowship  Mathematical Analysis of Strongly Correlated Processes on Discrete Dynamic Structures
Stauffer, A. (PI)
Engineering and Physical Sciences Research Council
1/04/16 → 30/09/22
Project: Research council