The Annals of Applied Probability

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

P. Caputo and A. Faggionato

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We consider random walks in a random environment which are generalized versions of well-known effective models for Mott variable-range hopping. We study the homogenized diffusion constant of the random walk in the one-dimensional case. We prove various estimates on the low-temperature behavior which confirm and extend previous work by physicists.

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Ann. Appl. Probab., Volume 19, Number 4 (2009), 1459-1494.

First available in Project Euclid: 27 July 2009

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Zentralblatt MATH identifier

Primary: 60K37: Processes in random environments 60G55: Point processes 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

Random walk in random environment point process invariance principle diffusion coefficient spectral gap isoperimetric constant


Caputo, P.; Faggionato, A. Diffusivity in one-dimensional generalized Mott variable-range hopping models. Ann. Appl. Probab. 19 (2009), no. 4, 1459--1494. doi:10.1214/08-AAP583.

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  • [1] Alexander, S. (1982). Variable range hopping in one-dimensional metals. Phys. Rev. B 26 2956.
  • [2] Ambegoakar, V., Halperin, B. I. and Langer, J. S. (1971). Hopping conductivity in disordered systems. Phys. Rev. B 4 2612–2620.
  • [3] Berger, N. and Biskup, M. (2007). Quenched invariance principle for simple random walk on percolation clusters. Probab. Theory Related Fields 137 83–120.
  • [4] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • [5] Breiman, L. (1953). Probability. Addison-Wesley, Reading, MA.
  • [6] Caputo, P. and Faggionato, A. (2007). Isoperimetric inequalities and mixing time for a random walk on a random point process. Ann. Appl. Probab. 17 1707–1744.
  • [7] Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.
  • [8] De Masi, A., Ferrari, P. A., Goldstein, S. and Wick, W. D. (1989). An invariance principle for reversible Markov processes. Applications to random motions in random environments. J. Statist. Phys. 55 787–855.
  • [9] Doyle, P. G. and Snell, J. L. (1984). Random Walks and Electric Networks. Carus Mathematical Monographs 22. Math. Assoc. Amer., Washington, DC.
  • [10] Ethier, S. N. and Kurz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [11] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Applications of Mathematics (New York) 33. Springer, Berlin.
  • [12] Faggionato, A. (2007). Bulk diffusion of 1D exclusion process with bond disorder. Markov Process. Related Fields 13 519–542.
  • [13] Faggionato, A. and Mathieu, P. (2008). Mott law for Mott variable-range random walk. Comm. Math. Phys. 281 263–286.
  • [14] Faggionato, A., Schulz-Baldes, H. and Spehner, D. (2006). Mott law as lower bound for a random walk in a random environment. Comm. Math. Phys. 263 21–64.
  • [15] Galambos, J. (1978). The Asymptotic Theory of Extreme Order Statistics. Wiley, New York.
  • [16] Kipnis, C. and Varadhan, S. R. S. (1986). Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104 1–19.
  • [17] Kurkijärvi, J. (1973). Hopping conductivity in one dimension. Phys. Rev. B 8 922–924.
  • [18] Ladieu, F. and Bouchaud, J.-P. (1993). Conductance statistics in small GaAs:Si wires at low temperatures. I. Theoretical analysis: Truncated fluctuations in insulating wires. J. Phys. I France 3 2311–2320.
  • [19] Lee, P. A. (1984). Variable range hopping in finite one-dimensional wires. Phys. Rev. Lett. 53 2042.
  • [20] Mathieu, P. and Piatnitski, A. (2007). Quenched invariance principles for random walks on percolation clusters. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463 2287–2307.
  • [21] Serota, R. A., Kalia, R. K. and Lee, P. A. (1986). New aspects of variable-range hopping in finite one-dimensional wires. Phys. Rev. B 33 8441.
  • [22] Shklovskii, B. and Efros, A. L. (1984). Electronic Properties of Doped Semiconductors. Springer, Berlin.
  • [23] Yu, Z. G. and Song, X. (2001). Variable range hopping and electric conductivity along the DNA double helix. Phys. Rev. Lett. 86 6018.
  • [24] Stone, C. (1963). Limit theorems for random walks, birth and death processes, and diffusion processes. Illinois J. Math. 7 638–660.