Localization properties of the anomalous diffusion phase in the directed trap model and in the Sinai diffusion with a bias.

Abstract

We study the localization properties of the anomalous diffusion phase x approximately t(micro) with 0<micro<1, which exists both in the Sinai diffusion with a small bias, and in the related directed trap model presenting a broad distribution of trapping times p(tau) approximately 1/tau(1+micro). Our starting point is the real space renormalization method, in which the whole thermal packet is considered to be in the same renormalized valley at large time: this assumption is asymptotically exact only in the limit of vanishing bias micro-->0 and corresponds to the Golosov localization. For finite micro we thus generalize the usual real space renormalization method to allow for the spreading of the thermal packet over many renormalized valleys. Our construction allows one to compute exact series expansions in micro for all observables: to compute observables at order micro (n), it is sufficient to consider in each sample a spreading of the thermal packet onto at most (1+n) traps. So our approach provides a description of the structure of the thermal packet sample by sample, and a full statistical characterization of the important traps at a given order in micro. For the directed trap model, we show explicitly up to order micro(2) how to recover the exact expressions for the diffusion front, the thermal width, and the localization parameter Y2. We then use our method to derive exact results for the localization parameters Y(k) for arbitrary k, the correlation function of two particles, and the generating function of thermal cumulants. We then explain how these results apply to the Sinai diffusion with bias by deriving the quantitative mapping between the large-scale renormalized descriptions of the two models. Finally we study the internal structure of the effective "traps" for the Sinai model via path-integral methods.