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.

中文翻译：

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在线上固定随机沙堆模型的统一阈值

我们考虑了阿贝尔随机沙堆模型。在此模型中，如果一个站点包含多个粒子，则该站点被视为不稳定。每个不稳定的位置都独立地以1的速率倾倒，将其两个粒子发送到独立选择的相邻位置。我们显示出，当初始平均密度小于1/2时，系统几乎可以肯定地进行本地固定。通过分析在沙堆动力学下大量颗粒访问每个站点的总次数的奇偶性，我们实现了这一界限。