无限体积沙堆模型的稳定性和渗流
Stabilizability and percolation in the infinite volume sandpile model
摘要：
在$\mathbb{Z}^d$上研究了无限体积中的沙堆模型。特别是，我们感兴趣的问题是否初始配置，选择根据一个固定的措施$μ$，是$μ$ -几乎肯定稳定。我们证明，稳定性不依赖于我们所采用的特定的稳定化程序。在$d=1$和$μ$的积测度与密度$ρ=1$（已知的临界值为稳定性在$d=1$）与一个积极的密度空站点，我们证明了$μ$是不稳定的。此外，我们研究，$ρ$值，使$μ$是稳定的，倾倒网站的渗流。我们发现，对于足够小的$ρ>0$，有一个亚临界制度，其中的一个集群的倾倒网站的分布有一个指数尾，是在亚临界制度的情况下，普通渗流。
英文摘要：
We study the sandpile model in infinite volume on $\mathbb{Z}^d$. In particular, we are interested in the question whether or not initial configurations, chosen according to a stationary measure $μ$, are $μ$-almost surely stabilizable. We prove that stabilizability does not depend on the particular procedure of stabilization we adopt. In $d=1$ and $μ$ a product measure with density $ρ=1$ (the known critical value for stabilizability in $d=1$) with a positive density of empty sites, we prove that $μ$ is not stabilizable. Furthermore, we study, for values of $ρ$ such that $μ$ is stabilizable, percolation of toppled sites. We find that for $ρ>0$ small enough, there is a subcritical regime where the distribution of a cluster of toppled sites has an exponential tail, as is the case in the subcritical regime for ordinary percolation.
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