Sinai model in presence of dilute absorbers

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.

21 pages, 2 figures

References in corpus (5)

• Strong disorder RG approach of random systems
• Transition from the annealed to the quenched asymptotics for a random walk on random obstacles
• One-dimensional classical diffusion in a random force field with weakly concentrated absorbers
• Breaking supersymmetry in a one-dimensional random Hamiltonian
• Distance traveled by random walkers before absorption in a random medium

Cited by in corpus (13)

• Strong Disorder RG approach - a short review of recent developments
• Perturbation Theory for Fractional Brownian Motion in Presence of Absorbing Boundaries
• One-dimensional disordered quantum mechanics and Sinai diffusion with random absorbers
• Supersymmetric quantum mechanics with Levy disorder in one dimension
• Effect of boundaries on the spectrum of a one-dimensional random mass Dirac Hamiltonian
• Random elastic networks : strong disorder renormalization approach
• Experimental test of Sinai's model in DNA unzipping
• Depinning and flow of a vortex line in an uniaxial random medium
• The generalized Lyapunov exponent for the one-dimensional Schrödinger equation with Cauchy disorder: some exact results
• Transport properties of diffusive particles conditioned to survive in trapping environments
• Anderson transition for Google matrix eigenstates
• Survival probability of random walks leaping over traps
• Ordered spectral statistics in 1D disordered supersymmetric quantum mechanics and Sinai diffusion with dilute absorbers