The Half-Plane Half-Comb walk is a random walk on the plane, when we have a square lattice on the upper half-plane and a comb structure on the lower half-plane, i.e., horizontal lines below the x-axis are removed. We prove that the probability that this walk returns to the origin in 2N steps is asymptotically equal to \(2/(\pi N).\) As a consequence, we prove strong laws and a limit distribution for the local time.

中文翻译：

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关于半平面半梳理行走的当地时间

Half-Plane Half-Comb行走是在平面上的随机行走，如果我们在上半个平面上有一个正方形格子，而在下半个平面上有一个梳状结构，即，x轴以下的水平线被删除了。我们证明了该步行以2 N步返回原点的概率渐近等于\（2 /（\ pi N）。\）。因此，我们证明了强大的定律和当地时间的极限分布。