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Suppose the points of two random walks in $\mathbb{R}^2$ are given the step number (or time) as a third coordinate, so that they become paths in $\mathbb{R}^3$. Here are several pairs of walks of $n=...

I am wondering if this random walk remains finite with positive probability. Start with three lines $A,B,C$ that are extensions of an equilateral triangle. Let $p_0$ be one corner. Generate a line $...