Abstract
The authors apply the replica method supplemented by a variational approach to directed paths which acquire random amplitudes and random phases. This problem is pertinent to the hopping of electrons in disordered materials in the variable-range-hopping regime. Here they consider the 'long-range' case in which both the amplitude and phase correlations of the disorder are long-ranged and a mixed case in which amplitude correlations are long-ranged and phase correlations are short-ranged. The variational approach is expected to become exact in the limit of infinite dimensions, and it yields the 'Flory' exponents for the transverse fluctuations. Stationary solutions which extremize the free energy and are characterized by non-trivial patterns of replica-symmetry breaking are obtained both analytically and numerically. The mixed case has a non-trivial phase diagram marked by abrupt changes in the pattern of replica-symmetry breaking. In this case the imaginary part of the random potential plays an important role.