Random Walks in Noninteger Dimension

TL;DR

This paper introduces a novel random walk model on concentric spheres that extends the concept of random walks to noninteger dimensions, providing valid probabilities and exact return probabilities involving the Riemann zeta function.

Contribution

It proposes a new random walk framework on a rotationally-symmetric geometry that is valid for all real dimensions, unlike traditional lattice-based models.

Findings

Exact return probability formula involving the Riemann zeta function

Valid probabilities for all real values of D

Number-theoretic interpretation of the results

Abstract

One can define a random walk on a hypercubic lattice in a space of integer dimension $D$. For such a process formulas can be derived that express the probability of certain events, such as the chance of returning to the origin after a given number of time steps. These formulas are physically meaningful for integer values of $D$. However, these formulas are unacceptable as probabilities when continued to noninteger $D$ because they give values that can be greater than $1$ or less than $0$. In this paper we propose a random walk which gives acceptable probabilities for all real values of $D$. This $D$-dimensional random walk is defined on a rotationally-symmetric geometry consisting of concentric spheres. We give the exact result for the probability of returning to the origin for all values of $D$ in terms of the Riemann zeta function. This result has a number-theoretic interpretation.