When a Random Walk of Fixed Length can Lead Uniformly Anywhere Inside a Hypersphere

A variation of the Pearson-Rayleigh random walk in which the steps are i.i.d. random vectors of exponential length and uniform orientation is considered. Conditioned on the total path length, the probability density function of the position of the walker after n steps is determined analytically in one and two dimensions. It is shown that in two dimensions n = 3 marks a critical transition point in the behavior of the random walk. By taking less than three steps and walking a total length l, one is more likely to end the walk near the boundary of the disc of radius l, while by taking more than three steps one is more likely to end near the origin. Somehow surprisingly, by taking exactly three steps one can end uniformly anywhere inside the disc of radius l. This means that conditioned on l, the sum of three vectors of exponential length and uniform direction has a uniform probability density. While the presented analytic approach provides a complete solution for all n, it becomes intractable in higher dimensions. In this case, it is shown that a necessary condition to have a uniform density in dimension d is that 2(d + 2)/d is an integer, equal to n + 1.