SPHERICALLY SYMMETRIC RANDOM WALKS I. REPRESENTATION IN TERMS OF ORTHOGONAL POLYNOMIALS

TL;DR

This paper introduces a polynomial-based framework to represent and analyze spherically symmetric random walks in arbitrary dimensions, enabling exact calculations of moments and Green's functions.

Contribution

It develops a novel polynomial representation for spherically symmetric random walks in any dimension, extending previous methods limited to integer dimensions.

Findings

Exact closed-form moments of the probability distribution are derived.

The polynomial approach facilitates calculation of the two-point Green's function.

The method applies to both classical random walks and quantum field theory contexts.

Abstract

Spherically symmetric random walks in arbitrary dimension $D$ can be described in terms of Gegenbauer (ultraspherical) polynomials. For example, Legendre polynomials can be used to represent the special case of two-dimensional spherically symmetric random walks. In general, there is a connection between orthogonal polynomials and semibounded one-dimensional random walks; such a random walk can be viewed as taking place on the set of integers $n$, $n=0,~1,~2,~\ldots$, that index the polynomials. This connection allows one to express random-walk probabilities as weighted inner products of the polynomials. The correspondence between polynomials and random walks is exploited here to construct and analyze spherically symmetric random walks in $D$-dimensional space, where $D$ is {\sl not} restricted to be an integer. The weighted inner-product representation is used to calculate exact closed-form spatial and temporal moments of the probability distribution associated with the random walk. The polynomial representation of spherically symmetric random walks is also used to calculate the two-point Green's function for a rotationally symmetric free scalar quantum field theory.