Random Walkers in 1-D Random Environments: Exact Renormalization Group Analysis

TL;DR

This paper provides an exact analytical study of Sinai's 1D diffusion model with random bias using renormalization group techniques, revealing detailed scaling laws, aging phenomena, and rare event statistics.

Contribution

It introduces an exact renormalization group analysis of Sinai's model, including effects of small bias, and derives comprehensive scaling laws and aging behaviors.

Findings

Exact scaling form of particle position distribution

Distribution of first passage times and meeting times

Aging regimes with novel logarithmic scaling

Abstract

Sinai's model of diffusion in one-dimension with random local bias is studied by a real space renormalization group which yields exact results at long times. The effects of an additional small uniform bias force are also studied. We obtain analytically the scaling form of the distribution of the position $x(t)$ of a particle, the probability of it not returning to the origin and the distributions of first passage times, in an infinite sample as well as in the presence of a boundary and in a finite size sample. We compute the distribution of meeting time of two particles. We also obtain a detailed analytic description of thermally averaged trajectories: we compute the distributions of the number of returns and of the number of jumps forward. They obey multifractal scaling, characterized by generalized persistence exponents $\theta(g)$ which we compute. With a small bias, the number of returns is finite, characterized by a universal scaling function. The statistics of the successive times of return of thermally averaged trajectories is obtained. The two time distribution of the positions of a particle, $x(t)$ and $x(t')$ ($t>t'$) is computed exactly. It exhibits ``aging'' with several regimes: without a bias, for $t-t' \sim t'^\alpha, \alpha > 1$, it exhibits a $(\ln t)/(\ln t')$ scaling, with a novel singularity at rescaled positions $x(t)=x(t')$. For closer times $\alpha<1$ there is a quasi-equilibrium regime with $\ln(t-t')/\ln t'$ scaling. The crossover to a $t/t'$ aging form under a small bias is obtained analytically. Rare events, e.g. splitting of the thermal packet between wells, are also studied. Connections with the Green's function of a 1D Schr\"odinger problem and quantum spin chains are discussed.