随机沙堆模型在线固定的统一阈值
Uniform threshold for fixation of the stochastic sandpile model on the line
摘要：
我们考虑阿贝尔随机沙堆模型。在这个模型中，当一个网站包含一个以上的粒子时，它被认为是不稳定的。每一个不稳定的位置都以$1$的速率独立地被推翻，将它的两个粒子送到独立选择的相邻位置。我们证明了当初始平均密度小于$1/2$时，系统几乎必然局部固定。我们通过分析沙堆动力学下大量粒子访问每个站点的总次数的奇偶性来实现这个界限。
英文摘要：
We consider the abelian stochastic sandpile model. In this model, a site is deemed unstable when it contains more than one particle. Each unstable site, independently, is toppled at rate $1$, sending two of its particles to neighbouring sites chosen independently. We show that when the initial average density is less than $1/2$, the system locally fixates almost surely. We achieve this bound by analysing the parity of the total number of times each site is visited by a large number of particles under the sandpile dynamics.
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