The Sinai model in the presence of dilute absorbers

Abstract

We study the Sinai model for the diffusion of a particle in aone-dimensional random potential in the presence of a small concentrationρ of perfect absorbers using the asymptotically exact real space renormalization method. Wecompute the survival probability, the averaged diffusion front and return probability, thetwo-particle meeting probability, the distribution of total distance traveled beforeabsorption and the averaged Green’s function of the associated Schrödinger operator. Ourwork confirms some recent results of Texier and Hagendorf obtained by Dyson–Schmidtmethods, and extends them to other observables and the presence of a drift. Inparticular the power law density of states is found to hold in all cases. Irrespective ofthe drift, the asymptotic rescaled diffusion front of surviving particles is foundto be a symmetric step distribution, uniform for , where ξ(t) is a new length scale for survival ( in the absence of drift). Survival outside this sharp region is found to decaywith a larger exponent, continuously varying with the rescaled distancex/ξ(t). A simple physical picture based on a saddle point is given, and universality isdiscussed.