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

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 $n$, $n=0,1,2,\dots$, 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.

31 pages, Revtex, uuencoded, (one ps-figure included)

References in corpus (8)

• Casimir effect for a $D$-dimensional sphere
• Dimensional Expansion for the Ising Limit of Quantum Field Theory
• Random Walks in Noninteger Dimension
• Spherically-Symmetric Random Walks in Noninteger Dimension
• Statistical Models on Spherical Geometries
• Polymer-Chain Adsorption Transition at a Cylindrical Boundary
• SPHERICALLY SYMMETRIC RANDOM WALKS III. POLYMER ADSORPTION AT A HYPERSPHERICAL BOUNDARY
• SPHERICALLY SYMMETRIC RANDOM WALKS II. DIMENSIONALLY DEPENDENT CRITICAL BEHAVIOR

Cited by in corpus (3)

• Wetting transitions for a random line in long-range potential
• SPHERICALLY SYMMETRIC RANDOM WALKS II. DIMENSIONALLY DEPENDENT CRITICAL BEHAVIOR
• SPHERICALLY SYMMETRIC RANDOM WALKS III. POLYMER ADSORPTION AT A HYPERSPHERICAL BOUNDARY