×

zbMATH — the first resource for mathematics

Scaling limit of sub-ballistic 1D random walk among biased conductances: a story of wells and walls. (English) Zbl 1441.60090
Summary: We consider a one-dimensional random walk among biased i.i.d. conductances, in the case where the random walk is transient but sub-ballistic: this occurs when the conductances have a heavy-tail at \(+\infty\) or at \(0\). We prove that the scaling limit of the process is the inverse of an \(\alpha\)-stable subordinator, which indicates an aging phenomenon, expressed in terms of the generalized arcsine law. In analogy with the case of an i.i.d. random environment studied in details in [N. Enriquez et al., Bull. Soc. Math. Fr. 137, No. 3, 423–452 (2009; Zbl 1186.60108); Ann. Appl. Probab. 23, No. 3, 1148–1187 (2013; Zbl 1279.60126)], some “traps” are responsible for the slowdown of the random walk. However, the phenomenology is somehow different (and richer) here. In particular, three types of traps may occur, depending on the fine properties of the tails of the conductances: (i) a very large conductance (a well in the potential); (ii) a very small conductance (a wall in the potential); (iii) the combination of a large conductance followed shortly after by a small conductance (a well-and-wall in the potential).
Reviewer: Reviewer (Berlin)
MSC:
60K37 Processes in random environments
60F17 Functional limit theorems; invariance principles
60G52 Stable stochastic processes
82D30 Statistical mechanical studies of random media, disordered materials (including liquid crystals and spin glasses)
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] [BABČ08] Gérard Ben Arous, Anton Bovier, and Jiří Černỳ, Universality of the REM for dynamics of mean-field spin glasses, Commun. Math. Phys. 282 (2008), no. 3, 663-695. · Zbl 1208.82024
[2] [BAČ08] Gérard Ben Arous and Jiří Černỳ, The arcsine law as a universal aging scheme for trap models, Commun. Pure Appl. Math. 61 (2008), no. 3, 289-329. · Zbl 1141.60075
[3] [BCKM98] Jean-Philippe Bouchaud, Leticia F. Cugliandolo, Jorge Kurchan, and Marc Mezard, Out of equilibrium dynamics in spin-glasses and other glassy systems, Spin glasses and random fields (1998), 161-223.
[4] [Ber96] Jean Bertoin, Lévy processes, vol. 121, Cambridge Tracts in Mathematics, 1996. · Zbl 0861.60003
[5] [Ber19] Quentin Berger, Notes on random walks in the Cauchy domain of attraction, Probab. Theory Relat. Fields 175 (2019), no. 1-2, 1-44. · Zbl 07109856
[6] [BGT89] Nicholas H. Bingham, Charles M. Goldie, and Jef L. Teugels, Regular variation, Encyclopedia of Mathematics and its applications, vol. 27, Cambridge University Press, 1989.
[7] [Bil68] Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, 1968.
[8] [Bis11] Marek Biskup, Recent progress on the random conductance model, Probab. Surv. 8 (2011), 294-373.
[9] [Bov06] Anton Bovier, Metastability: a potential theoretic approach, International Congress of Mathematicians. Vol. III, Eur. Math. Soc., Zürich, 2006, pp. 499-518. · Zbl 1099.60052
[10] [BS19] Quentin Berger and Michele Salvi, Scaling of sub-ballistic 1d random walks among biased random conductances, Markov Processes Relat. Fields 25 (2019), 171-187. · Zbl 07116988
[11] [Cli86] Daren B. H. Cline, Convolution tails, product tails and domains of attraction, Probab. Theory Relat. Fields 72 (1986), no. 4, 529-557. · Zbl 0577.60019
[12] [DG12] Dmitry Dolgopyat and Ilya Goldsheid, Quenched limit theorems for nearest neighbour random walks in 1d random environment, Commun. Math. Phys. 315 (2012), no. 1, 241-277. · Zbl 1260.60187
[13] [ESTZ13] Nathanaël Enriquez, Christophe Sabot, Laurent Tournier, and Olivier Zindy, Quenched limits for the fluctuations of transient random walks in random environment on \(\mathbb{Z}^1 \), Ann. Appl. Probab. 23 (2013), no. 3, 1148-1187. · Zbl 1279.60126
[14] [ESZ09a] Nathanaël Enriquez, Christophe Sabot, and Olivier Zindy, Aging and quenched localization for one-dimensional random walks in random environment in the sub-ballistic regime, Bull. Soc. Math. France 137 (2009), no. 3, 423-452. · Zbl 1186.60108
[15] [ESZ09b] Nathanaël Enriquez, Christophe Sabot, and Olivier Zindy, Limit laws for transient random walks in random environment on \(\mathbb{Z} \), Ann. Inst. Fourier (Grenoble) 59 (2009), no. 6, 2469-2508. · Zbl 1200.60093
[16] [FGS18] Alessandra Faggionato, Nina Gantert, and Michele Salvi, The velocity of 1d mott variable range hopping with external field, Ann. Inst. Henri Poincaré Probab. Statist. 54 (2018), no. 3, 1165-1203. · Zbl 1401.60182
[17] [FGS19] Alessandra Faggionato, Nina Gantert, and Michele Salvi, Einstein relation and linear response in one-dimensional mott variable-range hopping, Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 3, 1477-1508. · Zbl 07133728
[18] [FK18] Alexander Fribergh and Daniel Kious, Scaling limits for sub-ballistic biased random walks in random conductances, Ann. Probab. 46 (2018), no. 2, 605-686. · Zbl 1430.60089
[19] [FS19] Alessandra Faggionato and Michele Salvi, Regularity of biased 1D random walks in random environment, ALEA, Lat. Am. J. Probab. Math. Stat. 16 (2019), 1213-1248. · Zbl 1434.60308
[20] [Gol86] Andrey O. Golosov, Limit distributions for random walks in random environments, Soviet Math. Dokl. 28 (1986), 18-22.
[21] [Igl72] Donald L. Iglehart, Extreme values in the GI/G/1 queue, Ann. Math. Stat. 43 (1972), no. 2, 627-635. · Zbl 0238.60072
[22] [Kes86] Harry Kesten, The limit distribution of sinai’s random walk in random environment, Phys. A: Stati. Mech. Appl. 138 (1986), no. 1-2, 299-309. · Zbl 0666.60065
[23] [KKS75] Harry Kesten, Mykyta V Kozlov, and Frank Spitzer, A limit law for random walk in a random environment, Compositio Math. 30 (1975), 145-168. · Zbl 0388.60069
[24] [LD16] Hoang-Chuong Lam and Jerôme Depauw, Einstein relation for reversible random walks in random environment on \(\mathbb{Z} \), Stoch. Processes Appl. 126 (2016), no. 4, 983-996. · Zbl 1335.60188
[25] [LP16] Russell Lyons and Yuval Peres, Probability on trees and networks, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 42, Cambridge University Press, New York, 2016. · Zbl 1376.05002
[26] [Nag79] Serguey V. Nagaev, Large deviations of sums of independent random variables, Ann. Probab. 7 (1979), no. 5, 745-789. · Zbl 0418.60033
[27] \([NJB^+10]\) A. V. Nenashev, F. Jansson, S. D. Baranovskii, R. Österbacka, A. V. Dvurechenskii, and F. Gebhard, Effect of electric field on diffusion in disordered materials. I. one-dimensional hopping transport, Phys. Rev. B 81 (2010), no. 11, 115203. · Zbl 1180.82103
[28] [PS13] Jonathon Peterson and Gennady Samorodnitsky, Weak quenched limiting distributions for transient one-dimensional random walk in a random environment, Ann. Inst. Henri Poincaré Probab. Stat. 49 (2013), no. 3, 722-752. · Zbl 1277.60188
[29] [Sei00] Paul Seignourel, Discrete schemes for processes in random media, Probab. Theory Relat. Fields 118 (2000), no. 3, 293-322. · Zbl 0968.60100
[30] [Sin83] Ya G. Sinaĭ, The limiting behavior of a one-dimensional random walk in a random medium, Theory Probab. Appl. 27 (1983), no. 2, 256-268.
[31] [Sol75] Fred Solomon, Random walks in a random environment, Ann. Probab. 3 (1975), no. 1, 1-31. · Zbl 0305.60029
[32] [Whi02] Ward Whitt, Stochastic-process limits: an introduction to stochastic-process limits and their application to queues, Springer Science & Business Media, 2002. · Zbl 0993.60001
[33] [Zei04] Ofer Zeitouni, Random walks in random environment, Lectures on probability theory and statistics, Lecture Notes in Math., vol. 1837, Springer, Berlin, 2004, pp. 189-312. · Zbl 1060.60103
[34] [Zin09] Olivier Zindy, Scaling limit and aging for directed trap models, Markov Process. Relat. Fields 15 (2009), no. 1, 31-50. · Zbl 1177.60077
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.