Active phase for activated random walks on the lattice in all dimensions
Active phase for activated random walks on the lattice in all dimensions
复制标题
在所有维度的晶格上激活随机游走的活动阶段
DOI：
10.1214/22-aihp1341
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发表时间：
2022
期刊：
影响因子：
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通讯作者：
A. Gaudilliere
中科院分区：
文献类型：
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作者：
Nicolas Forien;A. Gaudilliere
我们证明了当睡眠率$\lambda$足够小时，$\mathbb{Z}^d$上的激活随机游走模型的临界密度严格小于1，当$\lambda\to 0$时，在任意维数$d\geqslant 1$上，临界密度趋于0 $.据我们所知，对于$d=2$，结果是新的。我们证明了这一点，显示，足够高的密度和足够小的睡眠率，稳定时间的$d$维环面模型是指数大。为了做到这一点，我们固定了粒子最终入睡的位置，这将问题简化为一个更简单的密度模型。利用该模型的阿贝尔性质，我们证明了具有负漂移的一维随机游动的稳定时间随机支配逃逸时间。然后，我们检查，这慢相的有限体积动力学意味着存在一个活跃的阶段上的无限晶格。
We show that the critical density of the Activated Random Walk model on $\mathbb{Z}^d$ is strictly less than one when the sleep rate $\lambda$ is small enough, and tends to $0$ when $\lambda\to 0$, in any dimension $d\geqslant 1$. As far as we know, the result is new for $d=2$. We prove this by showing that, for high enough density and small enough sleep rate, the stabilization time of the model on the $d$-dimensional torus is exponentially large. To do so, we fix the the set of sites where the particles eventually fall asleep, which reduces the problem to a simpler model with density one. Taking advantage of the Abelian property of the model, we show that the stabilization time stochastically dominates the escape time of a one-dimensional random walk with a negative drift. We then check that this slow phase for the finite volume dynamics implies the existence of an active phase on the infinite lattice.