SPHERICALLY SYMMETRICAL RANDOM-WALKS .1. REPRESENTATION IN TERMS OF ORTHOGONAL POLYNOMIALS

It is shown that, in general, a connection exists between orthogonal p olynomials and semibounded random walks. This connection allows one to view a random walk as taking place on the set of integers that index the orthogonal polynomials. An illustration is provided by the case of spherically symmetric random walks. The correspondence between orthog onal polynomials and random walks enables one to express random-walk p robabilities as weighted inner products of the polynomials. This corre spondence is exploited to construct and analyze spherically symmetric random walks in D-dimensional space, where D is not restricted to be a n integer. Such random walks can be described in terms of Gegenbauer ( ultraspherical) polynomials. For example, Legendre polynomials can be used to represent !he special case of two-dimensional spherically sym metric random walks. The weighted inner-product representation is used to calculate exact closed-form spatial and temporal moments of the pr obability distribution associated with the random walk. The polynomial representation of spherically symmetric random walks is then used to calculate the two-point Green's function for a rotationally symmetric free scalar quantum field theory.