| How Likely Is Pólya’s Drunkard to Return to the Pub Without Getting Mugged? (In d-Dimensional Manhattan [d ≥ 2]) (2008) | |||||||||||||||
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| In 1921, George Pólya[P] (see also [DS], ch. 7) famously proved that if a drunkard leaves a pub situated at the origin, in a d-dimensional Manhattan (w/o Broadway), and where he is allowed to only walk on streets and avenues, then he will return to the pub with probability 1 if d ≤ 2, but with probability < 1 if d> 2. Later writers (see [F], sec. 5.9) computed (calling this probability pd), that p3 =.340537..., p4 =.193201..., p5 =.135178..., p6 =.104715..., etc. But Pólya unrealistically assumed that it is safe to walk everywhere in Manhattan. As we all know, in some areas, a person-especially if he looks drunk-is guaranteed to get mugged. In 1989, Jet Wimp and I ([WimZ]) determined that, in a 3-dimensional Manhattan, where it is only safe to walk in {(x1, x2, x3) ∈ Z 3 | x1 ≥ x2 ≥ x3}, the probability of returning to the pub, without getting mugged, is.0648447.... But what about other dimensions and other regions? In this note, we will answer the following questions. What are the probabilities of a simple random walker, starting at the origin of Z d, and walking with unit positive steps, to return to the origin and stay in: (a) {(x1,..., xd) ∈ Z d | x1 ≥ 0, x2 ≥ 0,..., xd ≥ 0}, (b) {(x1,..., xd) ∈ Z d | x1 ≥... ≥ xd}, (c) {(x1,..., xd) ∈ Z d | x1 ≥... ≥ xd ≥ 0}? Calling these quantities ad, bd, cd, we found, using the Maple package DRUNKARD accompanying this article, that | |||||||||||||||
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