Lectures on Directed Paths in Random Media

Many physical problems involve calculating sums over paths. Each path could represent one possible physical realization of an object such as a polymer, in which case the weight of the path is the probability of that configuration. The weights themselves could be imaginary as in the case of Feynman paths describing the amplitude for the propagation of a particle. Path integral calculations are now a standard tool of the theoretical physicist, with many excellent books devoted to the subject [1,2]. What happens to sums over paths in the presence of quenched disorder in the medium? Individual paths are no longer weighted simply by their length, but are influenced by the impurities along their route. The sum may be dominated by " optimal " paths pinned to the impurities; the optimal paths usually forming complex hierarchical structures. Physical examples are provided by the interface of the random bond Ising model in two dimensions, and by magnetic flux lines in superconductors. The actual value of the sum naturally depends on the particular realization of randomness and varies from sample to sample. I shall initially describe the problem in the context of the high temperature expansion for the random bond Ising model. Introducing the sums over paths for such a lattice model avoids the difficulties associated with short distance cutoffs. Furthermore, the Ising model is sufficiently well understood to make the nature of various approximations more evident. The high temperature correlation functions of the Ising model are dominated by the shortest paths connecting the spins. Such configurations, that exclude loops and overhangs, are referred to as directed paths. They dominate the asymptotic behavior of the sum over distances that are much longer than the correlation length. Most of the lectures are devoted to describing the statistical properties of sums over such directed paths. As in all multiplicative noise processes, the probability distribution for the sum is broad. Hence Monte Carlo simulations may not be an appropriate tool for numerical studies; failing to find typical members of the ensemble. Instead, we shall present a transfer matrix method that allows a numerical evaluation of the sum in polynomial time in the length of the path. The results indeed show that the sum has a broad probability distribution that resembles (but is not quite) log–normal. To obtain analytical information about this probability distribution we shall introduce the replica method for examining the moments. A brief review …