Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

The velocity of 1d Mott variable-range hopping with external field

Alessandra Faggionato, Nina Gantert, and Michele Salvi

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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 (zero linear speed), and the existence in the ballistic regime of an invariant distribution for the environment viewed from the walker, which is mutually absolutely continuous with respect to the original law of the environment. If the point process is a renewal process, the aforementioned conditions result in a sharp criterion for ballisticity. Interestingly, the speed is not always continuous as a function of the bias.


Le « Mott variable-range hopping » est un mécanisme décrivant la conduction des electrons dans des solides désordonnés dans le régime de localisation d’Anderson. Sous l’approximation de champ moyen, le modèle se réduit à une marche aléatoire (marche aléatoire de Mott) sur un processus ponctuel. Cette marche peut sauter d’un point du processus ponctuel à n’importe quel autre, les sauts ne sont donc pas limités en taille.

Nous considerons une marche aléatoire de Mott unidimensionelle soumis à un champ extérieur (équivalent à un biais à droite). Nous montrons que la marche biaisée est transiente, et nous étudions sa vitesse linéaire. Nos résultats principaux sont des conditions pour la ballisticité (vitesse strictement positif) et la sous-ballisticité (vitesse nulle). Dans le regime ballistique, nous montrons l’existence d’une mesure invariante pour l’environment vu par la particule, absolument continue par rapport à la mesure originale. Si le processus ponctuel est un processus de renouvellement, nos conditions deviennent une condition nécessaire et suffisante pour la ballisticité. Nous montrons ainsi que la vitesse de la marche n’est pas, en général, une fonction continue du biais.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 3 (2018), 1165-1203.

Received: 11 May 2016
Revised: 6 March 2017
Accepted: 3 April 2017
First available in Project Euclid: 11 July 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K37: Processes in random environments 82D30: Random media, disordered materials (including liquid crystals and spin glasses) 60G50: Sums of independent random variables; random walks 60G55: Point processes

Random walk in random environment Disordered media Ballisticity Environment viewed from the walker Electron transport in disordered solids


Faggionato, Alessandra; Gantert, Nina; Salvi, Michele. The velocity of 1d Mott variable-range hopping with external field. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 3, 1165--1203. doi:10.1214/17-AIHP836.

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