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ABSTRACT:
We apply the replica method supplemented by a variational aproach 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 we 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 tranverse 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.
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