Directed paths with random phases

Directed Feynman paths in 1 + 1 dimensions that acquire random phases are examined numerically and analytically. This problem is relevant for the behavior of the conductance in two-dimensional amorphous insulators in the variable-range-hopping regime. Large-scale numerical simulations were performed on a model with short-range correlations. For the scaling of the transverse fluctuations ( t^ν), we obtain ν = 0.68 ± 0.025; and for the r.m.s free-energy fluctuations ( t^ω), we obtain ω = 0.335 ± 0.01. Up to 100 000 random samples were used for times as large as 2000. These results seem to exclude a recent conjecture that ν = 3/4 and ω = 1/2. Two versions of a model with long-range correlations are solved and shown to yield ν = 1/2; a physical explanation is given.