可分沙堆的定量爆炸与渗流
Quantitative explosion and percolation of the divisible sandpile
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中文总结 AI 辅助
针对Zd上的可分沙堆模型，在Levine等人证明均值为1时该过程会爆炸的基础上，本文确定了不同维度下某格点在时间t前释放质量的量级，还证明了对所有d≥2，该模型的 toppling 格点集在均值低于1时含无限连通分支，存在非平凡渗流相变，回答了相关问题。
中文摘要 AI 辅助
$\boldsymbol{Z}^d$上的可分沙堆从每个格点的独立同分布质量开始，在每个离散时间步长中，质量超过1的格点保留1个单位，并将多余部分平均发送给其邻居。Levine、Murugan、Peres和Ugurcan（2016）证明，在均值为1时，该过程会发生爆炸，每个格点都释放出无限质量。我们证明，对于$d\leq3$，某格点在时间$t$前释放的质量量级为$t^{(4-d)/4}$；对于$d=4$，量级为$\\
英文摘要
The divisible sandpile on $\mathbb{Z}^d$ starts from i.i.d. masses at each site, and, in each discrete time step, a site with mass above one keeps one unit and sends the excess equally to its neighbors. Levine, Murugan, Peres and Ugurcan (2016) showed that at mean one this process explodes, with every site emitting infinite mass. We show that the mass emitted from a site by time $t$ is of order $t^{(4-d)/4}$ for $d\leq3$, of order $\log t$ for $d=4$, and a tail-dependent, divergent rate for $d\geq5$. We further show that the mass emitted, after diffusive rescaling, converges to a Brownian optimal-stopping value for $d\leq3$ and to tail-dependent, weighted membrane fields for $d\geq5$, while at the critical dimension $d=4$, after superdiffusive rescaling, it converges to the membrane model. Using these estimates, we prove that, for every $d\geq2$, the set of sites that topple contains an infinite component at some mean below one, hence it has a non-trivial percolation phase transition. This answers a variant of a question of Fey, Meester and Redig (2009). The proof adapts ideas from the theory of level-set percolation of strongly correlated Gaussian fields.
