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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 
ASJC Scopus subject areas
 Mathematics(all)
 Applied Mathematics
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Alexandre Stauffer
Person: Research & Teaching