2004/01/13 by Jean-Michel Caillol · 20 citations
Mathematics · Physics and Astronomy · #Combinatorics #Euclidean space #Geometry #Hypersphere #Mathematical analysis #Mathematical physics #Mathematics #Physics #Polynomial #Random walk #Scientific Research and Discoveries #Space (punctuation) #Stochastic processes and statistical mechanics #Surface (topology) #Theoretical and Computational Physics #cond-mat.stat-mech
paper · pdf · doi:10.1088/0305-4470/37/9/001
published in Journal of Physics A Mathematical and General 37(9), 3077-3083 (Institute of Physics) · 10 pages
arxiv created 2004/01/13 · openalex publication_date 2004/02/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We consider random walks on the surface of the sphere S n −1 ( n ⩾ 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 ⟩ ∝ t valid in E n −1 should be replaced in S n −1 by the generic law ⟨cos θ⟩ ∝ exp(− t /τ), where θ denotes the angular displacement of the walker. More generally one has ⟨ C n /2−1 L (cos θ)⟩ ∝ exp(− t /τ( L , n )) where C n /2−1 L is a Gegenbauer polynomial. Conjectures concerning random walks on a fractal inscribed in S n −1 are given tentatively.