论文标题
水平垂直步行的复发
Recurrence of horizontal-vertical walks
论文作者
论文摘要
考虑在二维整数晶格上最近的邻居随机行走，在该晶格中，每个顶点最初都均匀和独立地标记为“ H”或“ V”。在每个离散的时间步骤中，Walker在当前位置重新示例标签（将h'更改为`v'和`v'和`v'to <h'，概率$ q $）。然后，如果新标签为“ h”，则采用平均零水平步骤，如果新标签为“ V”，则平均为零垂直步骤。该模型是确定性转子步行的随机版本，它的复发（即，经常访问每个顶点的概率1）在两个维度上仍然是一个开放的问题。我们通过证明水平垂直步行是$ q \ in（\ frac {1} {3}，1] $的$ q \，我们回答了水平垂直步行的类似问题。
Consider a nearest neighbor random walk on the two-dimensional integer lattice, where each vertex is initially labeled either `H' or `V', uniformly and independently. At each discrete time step, the walker resamples the label at its current location (changing `H' to `V' and `V' to `H' with probability $q$). Then, it takes a mean zero horizontal step if the new label is `H', and a mean zero vertical step if the new label is `V'. This model is a randomized version of the deterministic rotor walk, for which its recurrence (i.e., visiting every vertex infinitely often with probability 1) in two dimensions is still an open problem. We answer the analogous question for the the horizontal-vertical walk, by showing that the horizontal-vertical walk is recurrent for $q \in (\frac{1}{3},1]$.
