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

## TL;DR

This paper demonstrates that the Abelian oil and water model, unlike similar systems, does not exhibit an absorbing-state phase transition and remains in a fixation regime regardless of particle density.

## Contribution

It establishes that the oil and water model fundamentally differs from sandpile and activated random walk models by lacking an absorbing-state phase transition.

## Key findings

- The oil and water model does not undergo an absorbing-state phase transition.
- The model is in the fixation regime at all densities.
- This behavior holds for any vertex transitive graph and broad initial configurations.

## 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.