Sinai model in presence of dilute absorbers

TL;DR

This paper analyzes the Sinai model with dilute absorbers using real space renormalization, computing various observables and confirming recent results while extending them to include drift effects and new observables.

Contribution

It provides asymptotically exact calculations of multiple observables in the Sinai model with absorbers, extending previous Dyson-Schmidt results and analyzing the impact of drift.

Findings

Survival probability and diffusion front are characterized by a new survival length scale.

The asymptotic diffusion front is a symmetric step distribution within a specific region.

Survival probability outside this region decays with a larger, continuously varying exponent.

Abstract

We study the Sinai model for the diffusion of a particle in a one dimension random potential in presence of a small concentration $\rho$ of perfect absorbers using the asymptotically exact real space renormalization method. We compute the survival probability, the averaged diffusion front and return probability, the two particle meeting probability, the distribution of total distance traveled before absorption and the averaged Green's function of the associated Schrodinger operator. Our work confirms some recent results of Texier and Hagendorf obtained by Dyson-Schmidt methods, and extends them to other observables and in presence of a drift. In particular the power law density of states is found to hold in all cases. Irrespective of the drift, the asymptotic rescaled diffusion front of surviving particles is found to be a symmetric step distribution, uniform for $|x| < {1/2} \xi(t)$, where $\xi(t)$ is a new, survival length scale ($\xi(t)=T \ln t/\sqrt{\rho}$ in the absence of drift). Survival outside this sharp region is found to decay with a larger exponent, continuously varying with the rescaled distance $x/\xi(t)$. A simple physical picture based on a saddle point is given, and universality is discussed.