Random walkers in one-dimensional random environments: Exact renormalization group analysis

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 but large sample. We compute the distribution of meeting time of two particles in the same environment. We also obtain a detailed analytic description of the thermally averaged trajectories by computing quantities such as the joint distribution of the number of returns and of the number of jumps forward. These quantities obey multifractal scaling, characterized by generalized persistence exponents θ(g) which we compute. In the presence of a small bias, the number of returns to the origin becomes finite, characterized by a universal scaling function which we obtain. The full statistics of the distribution of successive times of return of thermally averaged trajectories is obtained, as well as detailed analytical information about correlations between directions and times of successive jumps. The two time distribution of the positions of a particle, x(t) and x(t ′ ) with t > t ′ , is also computed exactly. It is found to exhibit "aging" with several time regimes characterized by different behaviors. In the unbiased case, for t − t ′ ∼ t ′α with α > 1, it exhibits a ln t ln t ′ scaling, with a singularity at coinciding rescaled positions x(t) = x(t ′ ). This singularity is a novel feature, and corresponds to particles which remain in a renormalized valley. For closer times α < 1, the two time diffusion front exhibits a quasi-equilibrium regime with a ln(t − t ′ )/ ln t ′ behavior which we compute. The crossover to 1 a t/t ′ aging form in the presence of a small bias is also obtained analytically. Rare events corresponding to intermittent splitting of the thermal packet between separated wells which dominate some averaged observables, are also characterized in detail. Connections with the Green's function of a one-dimensional Schrödinger problem and quantum spin chains are discussed.

Scaling Green-Kubo Relation and Application to Three Aging Systems

“…(12), the overall value of the correlation function also increases with the age of the system. A similar type of correlation function, although with a logarithmic time dependence of the prefactor, has been found for a random walker in a random environment [41]. Summarizing, we thus have four different classes of correlation functions: Stationary, integrable correlation functions are covered by the usual Green-Kubo formula Eq.…”

Aging and quenched localization for one-dimensional random walks in random environment in the sub-ballistic regime

“…Now, the key observation is that with a probability converging to 1, the particle at time t is located at the foot of a valley having depth and Let us finally mention that Theorem 1 generalizes the aging result obtained by heuristic methods of renormalization by Le Doussal, Fisher and Monthus in [23] in the limit case when the bias of the random walk defining the potential tends to 0 (the case when this bias is 0 corresponding to the recurrent regime for the random walk in random environment). The recurrent case, which also leads to aging phenomenon, was treated in the same article and rigorous arguments were later presented by Dembo, Guionnet and Zeitouni in [12].…”

A Numerical Approach to Copolymers at Selective Interfaces

“…[27]) are set forth. The approaches are non rigorous, mostly based on replica computations, with the exception of [20] whose method is the real space renormalization technique for one-dimensional disordered systems first proposed in [13] in the context of quantum Ising model with transverse field and then applied with remarkably precise results to random walk in random environment, see e. g. [17]. The result in [3], that h(·) ≤ h c (·), is obtained by exploiting the path behavior of the copolymer near criticality suggested in [20].…”