Random walks in a disordered system. Representation of the Green's function by a functional intergral

Abstract

We consider the problem of describing long-time asymptotics in the problem of random walks in a statistically disordered system. Representations of the propagator are constructed with functional integrals in which averaging over the spatial distribution of impurity is derived in explicit form. In the limit of a continuous medium, representations of “field-theoretical” type lead to strong ultraviolet divergences which are not in “quantum-mechanical” variants of the theory (QMT). Within the framework of QMT we analyze the contribution of the trivial extremal with constant velocity and small constant momentum, and discover the existence of nontrivial extremals.