圆柱图上的内部DLA：波动和混合
Internal DLA on cylinder graphs: fluctuations and mixing
摘要：
我们使用耦合的思想引入\cite{levine2018long}表明，一个IDLA过程的圆柱图$G\times \mathbb{Z}$忘记一个典型的初始配置文件在$\mathcal{O}( N\sqrt{τ_N} (\log \! N)^2 )$步骤大$N$，其中$N$是基图$G$的大小，和$τ_N$是一个简单的随机游动在$G$的全变差混合时间。主要的新成分是IDLA在$G\times \mathbb{Z}$上的最大波动界，它只依赖于基图$G$的混合性质和阿贝尔性质。
英文摘要：
We use coupling ideas introduced in \cite{levine2018long} to show that an IDLA process on a cylinder graph $G\times \mathbb{Z}$ forgets a typical initial profile in $\mathcal{O}( N\sqrt{τ_N} (\log \! N)^2 )$ steps for large $N$, where $N$ is the size of the base graph $G$, and $τ_N$ is the total variation mixing time of a simple random walk on $G$. The main new ingredient is a maximal fluctuations bound for IDLA on $G\times \mathbb{Z}$ which only relies on the mixing properties of the base graph $G$ and the Abelian property.
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