Statistical mechanics of random paths on disordered lattices

Abstract

The dependence of the universality class on the statistical weight of unrestricted random paths is explicitly shown both for deterministic and statistical fractals such as the incipient infinite percolation cluster. Equally weighted paths (ideal chain) and kinetically generated paths (random walks) belong, in general, to different universality classes. For deterministic fractals exact renormalization group techniques are used. Asymptotic behaviors for the end-to-end distance ranging from power to logarithmic (localization) laws are observed for the ideal chain. In all these cases, random walks in the presence of nonperfect traps are shown to be in the same universality class of the ideal chain. Logarithmic behavior is reflected insingular renormalization group recursions. For the disordered case, numerical transfer matrix techniques are exploited on percolation clusters in two and three dimensions. The two-point correlation function scales with critical exponents not obeying standard scaling relations. The distribution of the number of chains and the number of chains returning to the starting point are found to be well approximated by a log-normal distribution. The logmoment of the number of chains is found to have an essential type of singularity consistent with the log-normal distribution. A non-self-averaging behavior is argued to occur on the basis of the results.