论文标题
激活随机步行的临界密度始终小于1
The Critical Density for Activated Random Walks is always less than 1
论文作者
论文摘要
激活的随机步行，在$ \ mathbb {z}^d $上，对于任何$ d \ geqslant 1 $，都是一种交互的粒子系统，在这种粒子系统中，粒子可以在两种状态中的任何一个：活动或冷冻。每个活动粒子在参数$λ$的指数时间内执行连续的时间简单随机步行，此后它仍保持在冷冻状态，直到另一个活性粒子共享其位置，并立即将其转移到活动中。众所周知，该模型具有相变，我们表明，控制相变的临界密度在任何维度上都小于一个，并且对于睡眠率$λ$的任何值。我们为小$λ$和大$λ$制度的临界密度提供上限。
Activated Random Walks, on $\mathbb{Z}^d$ for any $d\geqslant 1$, is an interacting particle system, where particles can be in either of two states: active or frozen. Each active particle performs a continuous-time simple random walk during an exponential time of parameter $λ$, after which it stays still in the frozen state, until another active particle shares its location, and turns it instantaneously back into activity. This model is known to have a phase transition, and we show that the critical density, controlling the phase transition, is less than one in any dimension and for any value of the sleep rate $λ$. We provide upper bounds for the critical density in both the small $λ$ and large $λ$ regimes.
