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

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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  • [1] V. Ambegoakar, B. I. Halperin and J. S. Langer. Hopping conductivity in disordered systems. Phys. Rev. B 4 (1971) 2612–2620.
  • [2] L. Avena, O. Blondel and A. Faggionato. Analysis of random walks in dynamic random environments via $L^{2}$-perturbations. Stochastic Process. Appl. 128 (2018) 3490–3530.
  • [3] G. Ben Arous, Y. Hu, S. Olla and O. Zeitouni. Einstein relation for biased random walk on Galton–Watson trees. Ann. Inst. Henri Poincaré Probab. Stat. 49 (2013) 698–721.
  • [4] A. Bovier. Metastability: A potential theoretic approach. In International Congress of Mathematicians. Vol. III 499–518. Eur. Math. Soc., Zürich, 2006.
  • [5] A. Bovier and F. den Hollander. Metastability: A Potential Theoretic Approach. Springer Verlag, Berlin, 2015.
  • [6] P. Caputo and A. Faggionato. Diffusivity in one-dimensional generalized Mott variable-range hopping models. Ann. Appl. Probab. 19 (2009) 1459–1494.
  • [7] F. Comets and S. Popov. Ballistic regime for random walks in random environment with unbounded jumps and Knudsen billiards. Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012) 721–744.
  • [8] D. J. Daley and D. Vere-Jones. An Introduction to the Theory of Point Processes. Springer Verlag, New York, 1988.
  • [9] A. De Masi, P. A. Ferrari, S. Goldstein and W. D. Wick. An invariance principle for reversible Markov processes. Applications to random motions in random environments. J. Stat. Phys. 55 (1989) 787–855.
  • [10] A. Faggionato, N. Gantert and M. Salvi. The velocity of 1d Mott variable range hopping with external field. Ann. Inst. Henri Poincaré Probab. Stat. 54 (2018) 1165–1203.
  • [11] A. Faggionato and M. Salvi. Regularity of biased 1D random walks in random environment. Preprint, 2018. Available at arXiv:1802.07874.
  • [12] A. Faggionato and F. Martinelli. Hydrodynamic limit of a disordered lattice gas. Probab. Theory Related Fields 127 (2003) 535–608.
  • [13] A. Faggionato and P. Mathieu. Mott law as upper bound for a random walk in a random environment. Comm. Math. Phys. 281 (2008) 263–286.
  • [14] A. Faggionato, H. Schulz-Baldes and D. Spehner. Mott law as lower bound for a random walk in a random environment. Comm. Math. Phys. 263 (2006) 21–64.
  • [15] P. Franken, D. König, U. Arndt and V. Schmidt. Queues and Point Processes. John Wiley and Sons, Chichester, 1982.
  • [16] N. Gantert, X. Guo and J. Nagel. Einstein relation and steady states for the random conductance model. Ann. Probab. 45 (2017) 2533–2567.
  • [17] N. Gantert, P. Mathieu and A. Piatnitski. Einstein relation for reversible diffusions in a random environment. Comm. Pure Appl. Math. 65 (2012) 187–228.
  • [18] X. Guo. Einstein relation for random walks in random environment. Ann. Probab. 44 (2016) 324–359.
  • [19] C. Kipnis and S. R. S. Varadhan. Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusion. Comm. Math. Phys. 104 (1986) 1–19.
  • [20] T. Komorowski, C. Landim and S. Olla. Fluctuations in Markov Processes. Grundlehren der Mathematischen Wissenschaften 345. Springer Verlag, Berlin, 2012.
  • [21] T. Komorowski and S. Olla. Einstein relation for random walks in random environments. Stochastic Process. Appl. 115 (2005) 1279–1301.
  • [22] T. Komorowski and S. Olla. On mobility and Einstein relation for tracers in time-mixing random environments. J. Stat. Phys. 118 (2005) 407–435.
  • [23] S. M. Kozlov. The method of averaging and walks in inhomogeneous environments. Russian Math. Surveys 40 (1985) 73–145.
  • [24] J. Kurkijärvi. Hopping conductivity in one dimension. Phys. Rev. B 8 (1973) 922–924.
  • [25] H.-C. Lam and J. Depauw. Einstein relation for reversible random walks in random environment on ${\mathbb{Z}}$. Stochastic Process. Appl. 126 (2016) 983–996.
  • [26] C. Landim, S. Olla and S. B. Volchan. Driven tracer particle in 1 dimensional symmetric simple exclusion. Comm. Math. Phys. 192 (1998) 287–307.
  • [27] J. L. Lebowitz and H. Rost. The Einstein relation for the displacement of a test particle in a random environment. Stochastic Process. Appl. 54 (1994) 183–196.
  • [28] M. Loulakis. Einstein relation for a tagged particle in simple exclusion processes. Comm. Math. Phys. 229 (2005) 347–367.
  • [29] M. Loulakis. Mobility and Einstein relation for a tagged particle in asymmetric mean zero random walk with simple exclusion. Ann. Inst. Henri Poincaré Probab. Stat. 41 (2005) 237–254.
  • [30] P. Maillard and O. Zeitouni. Performance of the Metropolis algorithm on a disordered tree: The Einstein relation. Ann. Appl. Probab. 24 (2014) 2070–2090.
  • [31] P. Mathieu and A. Piatnitski. Quenched invariance principles for random walks on percolation clusters. Proc. Roy. Soc. Edinburgh Sect. A 463 (2007) 2287–2307.
  • [32] P. Mathieu and A. Piatnitski. Steady states, fluctuation-dissipation theorems and homogenization for diffusions in a random environment with finite range of dependence. Available at arXiv:1601.02944.
  • [33] R. E. Megginson. An Introduction to Banach Space Theory. Springer Verlag, New York, 1998.
  • [34] A. Miller and E. Abrahams. Impurity conduction at low concentrations. Phys. Rev. 120 (1960) 745–755.
  • [35] N. F. Mott. On the transition to metallic conduction in semiconductors. Can. J. Phys. 34 (1956) 1356–1368.
  • [36] N. F. Mott. Conduction in glasses containing transition metal ions. J. Non-Crystal. Solids 1 (1968) 1–17.
  • [37] N. F. Mott. Conduction in non-crystalline materials III. Localized states in a pseudogap and near extremities of conduction and valence bands. Philos. Mag. 19 (1969) 835–852.
  • [38] N. F. Mott and E. A. Davis. Electronic Processes in Non-Crystaline Materials. Oxford University Press, New York, 1979.
  • [39] M. Pollak, M. Ortuño and A. Frydman. The Electron Glass. Cambridge University Press, Cambridge, 2013.
  • [40] J. Quastel. Bulk diffusion in a system with site disorder. Ann. Probab. 34 (2006) 1990–2036.
  • [41] M. Reed and B. Simon. Methods of Modern Mathematical Physics. Functional Analysis. Vol. I. Academic Press, San Diego, 1980.
  • [42] B. Shklovskii and A. L. Efros. Electronic Properties of Doped Semiconductors. Springer Verlag, Berlin, 1984.