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.

中文翻译：

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阿贝尔油水动力学没有吸收态相变

油水模型是一个相互作用的粒子系统，有两类粒子和一个粒子数守恒的动力学，属于所谓的阿贝尔网络类。此类中广泛研究的过程是沙堆模型和激活的随机游走，已知（至少对于底层图的某些选择）经历吸收状态相变。这种相变表征了两种状态的存在，这取决于粒子密度：低密度下的固定状态，其中动力学收敛于吸收状态并且每个粒子仅有限地跳跃多次，以及大密度下的活动状态，粒子无限频繁地跳跃，活动无限期地持续下去。在这项工作中，我们表明油和水模型与沙堆模型和激活的随机游走有很大不同，因为它不经历吸收状态相变并且在所有密度下都处于固定状态。我们的结果具有很大的普遍性：对于任何顶点传递的图和一大类初始配置。