Quantum coherence effects for strongly localized electrons

We examine the role of quantum coherence phenomena for strongly localized electrons. The probability distribution for tunneling between two sites separated by a distance t is computed numerically and analytically by summing all forward scattering paths. We find a universal probability distribution with mean proportional to t, its variance growing as t^2ω, where ω depends on dimension D. We find that two parameters are needed to describe the tunneling probability. We also study the response of the system to an external magnetic field B. A magnetic field (in the absence of spin orbit scattering) leads to an increased localization length ξ i.e. a positive magneto-conductance (MC). The above results are explained within a replica formalism from which we derive average behavior and fluctuations. Finally we analyze the critical hop directional averaging depending on disorder and the possibility of bulk MC anisotropy.