On The Range Of a Random Walk In A Torus
On The Range Of a Random Walk In A Torus
复制标题
环面随机游走的范围
DOI：
10.1214/14-aop924
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发表时间：
2010
影响因子：
0.8
通讯作者：
Eric Shellef
中科院分区：
文献类型：
--
作者：
Eviatar B. Procaccia;Eric Shellef
让一个简单的随机游走运行在三维或更高的环面的步骤数，这是一个恒定的体积比例。我们研究几何性质的范围，随机子图诱导的一组顶点访问的步行。距离和混合边界的典型范围内证明是一个$k$迭代的日志因子从那些在全环面为任意$k$。该证明使用了层次重整化和可能适用于欧几里得格中其他随机过程的技术。我们使用相同的技术来约束随机交错上的随机游走的热核。
Let a simple random walk run inside a torus of dimension three or higher for a number of steps which is a constant proportion of the volume. We examine geometric properties of the range, the random subgraph induced by the set of vertices visited by the walk. Distance and mixing bounds for the typical range are proven that are a $k$-iterated log factor from those on the full torus for arbitrary $k$. The proof uses hierarchical renormalization and techniques that can possibly be applied to other random processes in the Euclidean lattice. We use the same technique to bound the heat kernel of a random walk on random interlacements.