Diffusivity in One - Dimensional Generalized Mott Variable - Range Hopping Models

We consider random walks in a random environment which are generalized versions of well-known effective models for Mott variable-range hopping. We study the homogenized diffusion constant of the random walk in the one-dimensional case. We prove various estimates on the low-temperature behavior which confirm and extend previous work by physicists. 1. Introduction. Random walks among randomly distributed traps have been proposed as models to study the low-temperature behavior of conductivity in disordered solids in which the Fermi level (set equal to 0 below) lies in a region of strong Anderson localization. In the so-called Mott variable-range hopping model one considers trapping sites ξ = {x i } randomly distributed on R d , d ≥ 1, with a given density ρ. Each site x i is marked with a random energy E i ∈ [−1, 1], where the variables E i are independent and identically distributed according to some law ν on [−1, 1], and are assumed to be independent of ξ. The law ν satisfies ν[−E, E] ∼ |E| δ when E ≪ 1, for some positive constant δ. Then one considers a continuous-time random walk which starts at a given site x 0 and jumps from a site x i to any other site x j with rate