Spherically-Symmetric Random Walks in Noninteger Dimension

TL;DR

This paper investigates spherically-symmetric random walks in noninteger dimensions, providing asymptotic analysis, exact probability formulas, and intersection behavior, advancing understanding of random processes in fractional-dimensional spaces.

Contribution

It offers a detailed analysis of such random walks, including asymptotic behavior, exact probability calculations using Hurwitz functions, and intersection properties in noninteger dimensions.

Findings

Asymptotic behavior characterized for large times.

Exact probability formulas derived using Hurwitz functions.

Paths of multiple walkers rarely intersect in the continuum limit for certain dimensions.

Abstract

A previous paper (hep-lat/9311011) proposed a new kind of random walk on a spherically-symmetric lattice in arbitrary noninteger dimension $D$. Such a lattice avoids the problems associated with a hypercubic lattice in noninteger dimension. This paper examines the nature of spherically-symmetric random walks in detail. We perform a large-time asymptotic analysis of these random walks and use the results to determine the Hausdorff dimension of the process. We obtain exact results in terms of Hurwitz functions (incomplete zeta functions) for the probability of a walker going from one region of the spherical lattice to another. Finally, we show that the probability that the paths of $K$ independent random walkers will intersect vanishes in the continuum limit if $D> {{2K}\over{K-1}}$.