Diffusivity in one-dimensional generalized Mott variable-range hopping models

1D Mott variable-range hopping with external field

Mott variable-range hopping is a fundamental mechanism for electron transport in disordered solids in the regime of strong Anderson localization. We give a brief description of this mechanism, recall some results concerning the behavior of the conductivity at low temperature and describe in more detail recent results (obtained in collaboration with N. Gantert and M. Salvi) concerning the one-dimensional Mott variable-range hopping under an external field.

The velocity of 1D Mott variable range hopping with external field

Mott variable range hopping is a fundamental mechanism for low-temperature electron conduction in disordered solids in the regime of Anderson localization. In a mean field approximation, it reduces to a random walk (shortly, Mott random walk) on a random marked point process with possible long-range jumps. We consider here the one-dimensional Mott random walk and we add an external field (or a bias to the right). We show that the bias makes the walk transient, and investigate its linear speed. Our main results are conditions for ballisticity (positive linear speed) and for sub-ballisticity (ze…

Invariance principle for Mott variable range hopping and other walks on point processes

We consider a random walk on a homogeneous Poisson point process with energy marks. The jump rates decay exponentially in the A-power of the jump length and depend on the energy marks via a Boltzmann--like factor. The case A=1 corresponds to the phonon-induced Mott variable range hopping in disordered solids in the regime of strong Anderson localization. We prove that for almost every realization of the marked process, the diffusively rescaled random walk, with arbitrary start point, converges to a Brownian motion whose diffusion matrix is positive definite, and independent of the environment.…

一维Mott变程跳跃的反常标度区

Anomalous scaling regime for one-dimensional Mott variable-range hopping

该研究推导了一维Mott随机游走的反常次扩散标度极限，揭示了标度函数不连续点的阻挡机制，结合Bouchaud陷阱模型可融合阻挡与陷阱机制，证明基于过程收敛与电阻度量测度空间收敛的关联理论。

一维Mott变程跳跃的极值区域

Extremal regime for one-dimensional Mott variable-range hopping

该研究针对呈现严重阻塞的一维Mott随机游走的渐近行为展开，证明经适当重标度的游走位于两个环境可测势垒间且势垒位置有极值标度极限，同时给出了势垒间游走分布的渐近描述。

Einstein relation and linear response in one-dimensional Mott variable-range hopping

We consider one-dimensional Mott variable-range hopping with a bias, and prove the linear response as well as the Einstein relation, under an assumption on the exponential moments of the distances between neighboring points. In a previous paper \cite{FGS} we gave conditions on ballisticity, and proved that in the ballistic case the environment viewed from the particle approaches, for almost any initial environment, a given steady state which is absolutely continuous with respect to the original law of the environment. Here, we show that this bias--dependent steady state has a derivative at zer…