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Random walks, and in particular, their first passage times, are ubiquitous innature. Using direct enumeration of paths, we find the first return timedistribution of a 1D random walker, which is a heavy-tailed distribution withinfinite mean. Using the same method we find the last return time distribution,which follows the arcsine law. Both results have a broad range of applicationsin physics and other disciplines. The derivation presented here is readilyaccessible to physics undergraduates, and provides an elementary introductioninto random walks and their intriguing properties.

An elementary derivation of first and last return times of 1D random walks