Random Walks on Hyperspheres of Arbitrary Dimensions

Abstract

We consider random walks on the surface of the sphere $S_{n-1}$ ($n\ge 2$) of the $n$-dimensional Euclidean space $E_n$, in short a hypersphere. By solving the diffusion equation in $S_{n-1}$ we show that the usual law $<r^2>\propto t$ valid in $E_{n-1}$ should be replaced in $S_{n-1}$ by the generic law $<\cos \theta >\propto \exp (-t/\tau )$, where $\theta$ denotes the angular displacement of the walker. More generally one has $<C_L^{n/2-1}\cos (\theta )>\propto \exp (-t/\tau (L,n))$ where $C_L^{n/2-1}$ a Gegenbauer polynomial. Conjectures concerning random walks on a fractal inscribed in $S_{n-1}$ are given tentatively.

10 pages

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