Random Walks on Hyperspheres of Arbitrary Dimensions

TL;DR

This paper analyzes the behavior of random walks on hyperspheres of arbitrary dimensions, deriving new laws for angular displacement and exploring potential extensions to fractal structures.

Contribution

It provides analytical solutions for diffusion on hyperspheres and introduces generalized exponential decay laws involving Gegenbauer polynomials.

Findings

The mean squared displacement law is replaced by an exponential decay in angular correlation.

Derived explicit solutions for diffusion equations on $S_{n-1}$.

Proposes conjectures for random walks on fractals inscribed in hyperspheres.

Abstract

We consider random walks on the surface of the sphere $S_{n-1}$ ($n \geq 2$) 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 > \varpropto t $ valid in $E_{n-1}$ should be replaced in $S_{n-1}$ by the generic law $<\cos \theta > \varpropto \exp(-t/\tau)$, where $\theta$ denotes the angular displacement of the walker. More generally one has $<C^{n/2-1}_{L}\cos(\theta)> \varpropto \exp(-t/ \tau(L,n))$ where $C^{n/2-1}_{L}$ a Gegenbauer polynomial. Conjectures concerning random walks on a fractal inscribed in $S_{n-1}$ are given tentatively.