Extremal regime for one-dimensional Mott variable-range hopping
Annales Henri Lebesgue, Volume 6 (2023), pp. 1169-1211.

Metadata

Keywords random walk in random environment, disordered media, sub-diffusivity, Mott variable-range hopping, extremal process

Abstract

We study the asymptotic behaviour of a version of the one-dimensional Mott random walk in a regime that exhibits severe blocking. We establish that, for any fixed time, the appropriately-rescaled Mott random walk is situated between two environment-measurable barriers, the locations of which are shown to have an extremal scaling limit. Moreover, we give an asymptotic description of the distribution of the Mott random walk between the barriers that contain it.


References

[Ale82] Alexander, S. Variable range hopping in one-dimensional metals, Phys. Rev. B, Volume 26 (1982) no. 6, pp. 2956-2962 | DOI

[Bar98] Barlow, Martin T. Diffusions on fractals, Lectures on probability theory and statistics (Saint-Flour, 1995) (Lecture Notes in Mathematics), Volume 1690, Springer, 1998, pp. 1-121 | DOI | MR | Zbl

[Bar17] Barlow, Martin T. Random walks and heat kernels on graphs, London Mathematical Society Lecture Note Series, 438, Cambridge University Press, 2017 | DOI | MR | Zbl

[CF09] Caputo, Pietro; Faggionato, Alessandra Diffusivity in one-dimensional generalized Mott variable-range hopping models, Ann. Appl. Probab., Volume 19 (2009) no. 4, pp. 1459-1494 | MR | Zbl

[CFJ20] Croydon, David A.; Fukushima, Ryoki; Junk, Stefan Anomalous scaling regime for one-dimensional Mott variable-range hopping (2020) (preprint appears at arXiv:2010.01779)

[CFP13] Caputo, Pietro; Faggionato, Alessandra; Prescott, Timothy M. Invariance principle for Mott variable range hopping and other walks on point processes, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 49 (2013) no. 3, pp. 654-697 | Numdam | MR | Zbl

[CM15] Croydon, David A.; Muirhead, Stephen Functional limit theorems for the Bouchaud trap model with slowly varying traps, Stochastic Processes Appl., Volume 125 (2015) no. 5, pp. 1980-2009 | DOI | MR | Zbl

[CM17] Croydon, David A.; Muirhead, Stephen Quenched localisation in the Bouchaud trap model with slowly varying traps, Probab. Theory Relat. Fields, Volume 168 (2017) no. 1-2, pp. 269-315 | DOI | MR | Zbl

[CRR + 97] Chandra, Ashok K.; Raghavan, Prabhakar; Ruzzo, Walter L.; Smolensky, Roman; Tiwari, Prasoon The electrical resistance of a graph captures its commute and cover times, Comput. Complexity, Volume 6 (1997) no. 4, pp. 312-340 | DOI | MR | Zbl

[DS84] Doyle, Peter G.; Snell, J. Laurie Random walks and electric networks, Carus Mathematical Monographs, 22, Mathematical Association of America, 1984 | DOI | MR | Zbl

[FOT11] Fukushima, Masatoshi; Oshima, Yoichi; Takeda, Masayoshi Dirichlet forms and symmetric Markov processes, De Gruyter Studies in Mathematics, 19, Walter de Gruyter, 2011 | Zbl

[Gne43] Gnedenko, Boris Sur la distribution limite du terme maximum d’une série aléatoire, Ann. Math., Volume 44 (1943), pp. 423-453 | DOI | Zbl

[Kal02] Kallenberg, Olav Foundations of modern probability, Probability and Its Applications, Springer, 2002 | DOI | Zbl

[Kas86] Kasahara, Yuji A limit theorem for sums of i.i.d. random variables with slowly varying tail probability, J. Math. Kyoto Univ., Volume 26 (1986) no. 3, pp. 437-443 | MR | Zbl

[Kig12] Kigami, Jun Resistance forms, quasisymmetric maps and heat kernel estimates, 216, American Mathematical Society, 2012 | Zbl

[Lam64] Lamperti, John On extreme order statistics, Ann. Math. Stat., Volume 35 (1964), pp. 1726-1737 | DOI | MR | Zbl

[Lee84] Lee, Patrick A. Variable-range hopping in finite one-dimensional wires, Phys. Rev. Lett., Volume 53 (1984) no. 21, pp. 2042-2045 | DOI

[Mot69] Mott, Nevill F. Conduction in non-crystalline materials, Philos. Mag., Volume 19 (1969) no. 160, pp. 835-852 | DOI

[Mot72] Mott, Nevill F. Introductory talk; Conduction in non-crystalline materials, J. Non Cryst. Solids, Volume 8–10 (1972), pp. 1-18 (Amorphous and Liquid Semiconductors) | DOI

[Mui15] Muirhead, Stephen Two-site localisation in the Bouchaud trap model with slowly varying traps, Electron. Commun. Probab., Volume 20 (2015), 25 | DOI | MR | Zbl

[NP08] Nachmias, Asaf; Peres, Yuval Critical random graphs: diameter and mixing time, Ann. Probab., Volume 36 (2008) no. 4, pp. 1267-1286 | MR | Zbl

[Res87] Resnick, Sidney I. Extreme values, regular variation, and point processes, Applied Probability. A Series of the Applied Probability Trust, 4, Springer, 1987 | DOI | Zbl

[SKL86] Serota, R. A.; Kalia, R. K.; Lee, Patrick A. New aspects of variable-range hopping in finite one-dimensional wires, Phys. Rev. B, Volume 33 (1986) no. 12, pp. 8441-8446 | DOI

[SM98] Starykh, Oleg A.; Maslov, Dmitrii L. Conductance of a Mott Quantum Wire, Phys. Rev. Lett., Volume 80 (1998) no. 8, pp. 1694-1697 | DOI

[Whi02] Whitt, Ward Stochastic-process limits. An introduction to stochastic-process limits and their application to queues, Springer Series in Operations Research, Springer, 2002 | DOI