10 - Directed paths in random media

Summary

Introduction

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 [R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965)]; [F.W. Wiegel, Introduction to Path-Integral Methods in Physics and Polymer Science (World Scientific, Singapore, 1986)].

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 motivate 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.