Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

Alessandra Faggionato, Nina Gantert, and Michele Salvi

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We consider one-dimensional Mott variable-range hopping. This random walk is an effective model for the phonon-induced 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 $L^{p}$-bound, $p>2$, uniformly for small bias. This $L^{p}$-bound yields, by a general argument not involving our specific model, the statement about the linear response.


Nous considérons le modèle « Mott variable-range hopping ». Cette marche aléatoire décrit la conduction des electrons dans des solides désordonnés dans le régime de localisation forte d’Anderson lorsque la densité des porteurs de charge est faible. En particulier, nous considérons une marche aléatoire de Mott unidimensionelle soumise à un champ extérieur. Sous une hypothèse à propos des moments exponentiels de la distance entre les points consécutifs, nous montrons la réponse linéaire et la relation d’Einstein. Dans un travail précedent, voir (Ann. Inst. Henri Poincaré Probab. Stat. 54 (2018) 1165–1203), nous avons donné des conditions pour la ballisticité de la marche. En plus, nous avons montré que l’environnement vu de la particule converge en loi (pour presque tous les points de départ) vers une mesure invariante (état stationnaire) qui est absolument continue par rapport à la loi originale de l’environnement. Ici, nous montrons que cet état stationnaire a une dérivée en zéro par rapport au bias (réponse linéaire), et nous utilisons ce résultat pour démontrer la relation d’Einstein. Notre méthode est nouvelle : au lieu d’utiliser des arguments perturbatifs ou des temps de régéneration, nous donnons une borne en $L^{p}$, $p>2$, pour la densité de l’état stationnaire par rapport à la mesure invariante sans biais. Cette borne est uniforme dans le biais pour des biais qui sont proches de zéro. L’argument pour déduire la réponse linéaire de cette borne est général et ne dépend pas des détails de notre modèle.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 3 (2019), 1477-1508.

Received: 10 October 2017
Revised: 5 June 2018
Accepted: 20 August 2018
First available in Project Euclid: 25 September 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 60K37: Processes in random environments 60J25: Continuous-time Markov processes on general state spaces 60G55: Point processes 82D30: Random media, disordered materials (including liquid crystals and spin glasses)

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


Faggionato, Alessandra; Gantert, Nina; Salvi, Michele. Einstein relation and linear response in one-dimensional Mott variable-range hopping. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 3, 1477--1508. doi:10.1214/18-AIHP925.

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