The Sinai model in the presence of dilute absorbers

We study the Sinai model for the diffusion of a particle in a one-dimensional random potential in the presence of a small concentration. 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 the 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 vertical bar x vertical bar < (1)(2)xi(t), where xi(t) is a new length scale for survival (xi(t) = T ln t/root 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.