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

Alessandra Faggionato, Nina Gantert, Michele Salvi

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5 Scopus citations


We consider one-dimensional Mott variable-range hopping. This random walk is an effective model for the phononinduced hopping of electrons in disordered solids within the regime of strong Anderson localization at low carrier density. We introduce 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 (Ann. Inst. Henri Poincaré Probab. Stat. 54 (2018) 1165-1203) 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 zero in terms of the bias (linear response), and use this result to get the Einstein relation. Our approach is new: instead of using e.g. perturbation theory or regeneration times, we show that the Radon-Nikodym derivative of the bias-dependent steady state with respect to the equilibrium state in the unbiased case satisfies an Lp-bound, p > 2, uniformly for small bias. This Lp-bound yields, by a general argument not involving our specific model, the statement about the linear response.

Original languageEnglish
Pages (from-to)1477-1508
Number of pages32
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Issue number3
StatePublished - 2019


  • Einstein relation
  • Environment seen from the particle
  • Linear response
  • Mott variable-range hopping
  • Random conductance model
  • Random walk in random environment
  • Steady states


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