论文标题
在所有维度上在晶格上激活随机步行的活动阶段
Active Phase for Activated Random Walks on the Lattice in all Dimensions
论文作者
论文摘要
我们表明，当$ \ mathbb {z}^d $上激活的随机步行模型的临界密度严格少于睡眠率$λ$足够小时，并且在$λ\至0 $的任何尺寸$ d \ d \ d \ geqslant 1 $时倾向于$ 0 $。据我们所知，结果是$ d = 2 $的新结果。 我们通过表明，对于高度高和足够小的睡眠率，模型在$ d $维的圆环上的稳定时间呈指数较大。为此，我们修复了粒子最终入睡的一组站点，从而将问题降低到具有密度一个的简单模型。利用模型的阿贝尔特性，我们表明稳定时间随机地主导了一维随机行走的逃生时间，而阴性漂移。然后，我们检查有限体积动力学的缓慢相意味着无限晶格上存在活性相。
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 $λ$ is small enough, and tends to $0$ when $λ\to 0$, 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.
