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We can imagine simulating the random walk on an infinite line, keeping track of the "extension", which is the distance between the rightmost point visited and the leftmost point visited. Let $\ell(a,b)$ denote the probability that the extension became $b-a$ due to a move to $a$, and let $r(a,b)$ denote the probability that the extension became $b-a$ due to a ...


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