Competition between quenched disorder and long-range connections: A numerical study of diffusion

TL;DR

This study investigates how quenched disorder and long-range connections influence diffusion in one-dimensional random walks, revealing different regimes and critical behavior through numerical analysis.

Contribution

It provides a detailed numerical analysis of the interplay between quenched disorder and long-range connections at the critical point s=2, including bounds on the barrier exponent.

Findings

For s>2, long-range connections are irrelevant and diffusion is logarithmic.

For s<2, quenched disorder is irrelevant and diffusion is ballistic.

At s=2, a non-trivial barrier exponent governs activated scaling.

Abstract

The problem of random walk is considered in one dimension in the simultaneous presence of a quenched random force field and long-range connections the probability of which decays with the distance algebraically as p_l ~ \beta l^{-s}. The dynamics are studied mainly by a numerical strong disorder renormalization group method. According to the results, for s>2 the long-range connections are irrelevant and the mean-square displacement increases as <x^2(t)> ~ (ln t)^{2/\psi} with the barrier exponent \psi=1/2, which is known in one-dimensional random environments. For s<2, instead, the quenched disorder is found to be irrelevant and the dynamical exponent is z=1 like in a homogeneous environment. At the critical point, s=2, the interplay between quenched disorder and long-range connections results in activated scaling, however, with a non-trivial barrier exponent \psi(\beta), which decays continuously with \beta, but is independent of the form of the quenched disorder. Upper and lower bounds on \psi(\beta) are established and numerical estimates are given for various values of \beta. Beside random walks, accurate numerical estimates of the graph dimension and the resistance exponent are given for various values of \beta at s=2.