Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-26T13:14:49.469Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-Gaussian fluctuations of randomly trapped random walks

Part of: Limit theorems Stochastic processes Special processes Markov processes

Published online by Cambridge University Press:  08 October 2021

Adam Bowditch
Adam Bowditch*
Affiliation:
University College Dublin
*
*Postal address: University College Dublin, School of Mathematics and Statistics, Belfield, Dublin 4, Ireland. Email: adam.bowditch@ucd.ie
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we consider the one-dimensional, biased, randomly trapped random walk with infinite-variance trapping times. We prove sufficient conditions for the suitably scaled walk to converge to a transformation of a stable Lévy process. As our main motivation, we apply subsequential versions of our results to biased walks on subcritical Galton–Watson trees conditioned to survive. This confirms the correct order of the fluctuations of the walk around its speed for values of the bias that yield a non-Gaussian regime.

Keywords

Trap modelrandom environmentanomalous fluctuationsGalton–Watson treeLévy processesfunctional convergence

MSC classification

Primary: 60K37: Processes in random environments
Secondary: 60F17: Functional limit theorems; invariance principles 60G51: Processes with independent increments; L'evy processes 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 3 , September 2021 , pp. 801 - 838
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adékon, E. (2014). Speed of the biased random walk on a Galton–Watson tree. Prob. Theory Relat. Fields 159, 597–617.10.1007/s00440-013-0515-yCrossRefGoogle Scholar
Athreya, K. and Ney, P. (2004). Branching Processes. Dover, Mineola, NY.Google Scholar
Ben Arous, G., Cabezas, M., Černý, J. and Royfman, R. (2015). Randomly trapped random walks. Ann. Prob. 43, 2405–2457.CrossRefGoogle Scholar
Ben Arous, G. and Černỳ, J. (2006). Dynamics of trap models. In Mathematical Statistical Physics (Les Houches, Vol. 83), Elsevier, Amsterdam, pp. 331–394.10.1016/S0924-8099(06)80045-4CrossRefGoogle Scholar
Ben Arous, G. and Černỳ, J. (2007). Scaling limit for trap models on $\mathbb{Z}^d$. Ann. Prob. 35, 2356–2384.10.1214/009117907000000024CrossRefGoogle Scholar
Ben Arous, G. and Fribergh, A. (2016). Biased random walks on random graphs. In Probability and Statistical Physics in St. Petersburg (Proc. Symp. Pure Math. Vol. 91), American Mathematical Society, Providence, RI, pp. 99–153.Google Scholar
Ben Arous, G., Fribergh, A., Gantert, N. and Hammond, A. (2012). Biased random walks on Galton–Watson trees with leaves. Ann. Prob. 40, 280–338.CrossRefGoogle Scholar
Ben Arous, G. and Hammond, A. (2012). Randomly biased walks on subcritical trees. Commun. Pure Appl. Math. 65, 1481–1527.CrossRefGoogle Scholar
Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.10.1002/9780470316962CrossRefGoogle Scholar
Bolthausen, E. (1989). A central limit theorem for two-dimensional random walks in random sceneries. Ann. Prob. 17, 108–115.CrossRefGoogle Scholar
Bolthausen, E. and Sznitman, A. (2002). On the static and dynamic points of view for certain random walks in random environment. Methods Appl. Anal. 9, 345–375.Google Scholar
Bouchaud, J. (1992). Weak ergodicity breaking and aging in disordered systems. J. Physique I 2, 1705–1713.CrossRefGoogle Scholar
Bowditch, A. (2017). Biased randomly trapped random walks and applications to random walks on Galton–Watson trees. Doctoral Thesis, University of Warwick.Google Scholar
Bowditch, A. (2018). Escape regimes of biased random walks on Galton–Watson trees. Prob. Theory Relat. Fields 170, 685–768.CrossRefGoogle ScholarPubMed
Bowditch, A. (2018). A quenched central limit theorem for biased random walks on supercritical Galton–Watson trees. J. Appl. Prob. 55, 610–626.10.1017/jpr.2018.38CrossRefGoogle Scholar
Bowditch, A. (2019). Central limit theorems for biased randomly trapped walks on $\mathbb{Z}$. Stoch. Process. Appl. 129, 740–770.CrossRefGoogle Scholar
Černý, J. and Wassmer, T. (2015). Randomly trapped random walks on $\mathbb{Z}^d$. Stoch. Process. Appl. 125, 1032–1057.CrossRefGoogle Scholar
Croydon, D., Fribergh, A. and Kumagai, T. (2013). Biased random walk on critical Galton–Watson trees conditioned to survive. Prob. Theory Relat. Fields 157, 453–507.CrossRefGoogle Scholar
Dembo, A., Peres, Y. and Zeitouni, O. (1996). Tail estimates for one-dimensional random walk in random environment. Commun. Math. Phys. 181, 667–683.CrossRefGoogle Scholar
Dhar, D. (1984). Diffusion and drift on percolation networks in an external field. J. Phys. A 17, L257–L259.10.1088/0305-4470/17/5/007CrossRefGoogle Scholar
Feller, W. (1971). An Introduction to Probability Theory and its Applications, Vol. II, 2nd edn. John Wiley, New York.Google Scholar
Feng, X., Shao, Q., and Zeitouni, O.. (2019). Self-normalized moderate deviations for random walk in random scenery. J. Theoret. Prob. 34, 103–124.CrossRefGoogle Scholar
Fontes, L., Isopi, M. and Newman, C. (2002). Random walks with strongly inhomogeneous rates and singular diffusions: convergence, localization and aging in one dimension. Ann. Prob. 30, 579–604.CrossRefGoogle Scholar
Fontes, L. and Mathieu, P. (2014). On the dynamics of trap models in $\mathbb{Z}^d$. Proc. London Math. Soc. 108, 1562–1592.10.1112/plms/pdt064CrossRefGoogle Scholar
Fribergh, A. and Hammond, A. (2014). Phase transition for the speed of the biased random walk on the supercritical percolation cluster. Commun. Pure Appl. Math. 67, 173–245.CrossRefGoogle Scholar
Gantert, N., König, W. and Shi, Z. (2007). Annealed deviations of random walk in random scenery. Ann. Inst. H. Poincaré Prob. Statist. 43, 47–76.CrossRefGoogle Scholar
Geiger, J. and Kersting, G. (1999). The Galton–Watson tree conditioned on its height. In Probability Theory and Mathematical Statistics (Vilnius 1998), TEV, Vilnius, pp. 277–286.Google Scholar
Gnedenko, B. and Kolmogorov, A. (1954). Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Cambridge, MA.Google Scholar
Guillotin-Plantard, N., Hu, Y. and Schapira, B. (2013). The quenched limiting distributions of a one-dimensional random walk in random scenery. Electron. Commun. Prob. 18, 1–7.CrossRefGoogle Scholar
Hammond, A. (2013). Stable limit laws for randomly biased walks on supercritical trees. Ann. Prob. 41, 1694–1766.CrossRefGoogle Scholar
Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Kesten, H. (1986). Subdiffusive behavior of random walk on a random cluster. Ann. Inst. H. Poincaré Prob. Statist. 22, 425–487.Google Scholar
Kesten, H., Kozlov, M. and Spitzer, F. (1975). A limit law for random walk in a random environment. Compositio Math. 30, 145–168.Google Scholar
Kesten, H. and Spitzer, F. (1979). A limit theorem related to a new class of self-similar processes. Z. Wahrscheinlichkeitsth. 50, 5–25.CrossRefGoogle Scholar
Lyons, R., Pemantle, R. and Peres, Y. (1995). Conceptual proofs of $L\log L$ criteria for mean behavior of branching processes. Ann. Prob. 23, 1125–1138.Google Scholar
Lyons, R., Pemantle, R. and Peres, Y. (1996). Biased random walks on Galton–Watson trees. Prob. Theory Relat. Fields 106, 249–264.CrossRefGoogle Scholar
Lyons, R. and Peres, Y. (2016). Probability on trees and networks, volume 42 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, New YorkCrossRefGoogle Scholar
Meerschaert, M. and Scheffler, H. (2004). Limit theorems for continuous-time random walks with infinite mean waiting times. J. Appl. Prob. 41, 623–638.CrossRefGoogle Scholar
Mourrat, J. (2011). Scaling limit of the random walk among random traps on $\mathbb{Z}^d$. Ann. Inst. H. Poincaré Prob. Statist. 47, 813–849.CrossRefGoogle Scholar
Peres, Y. and Zeitouni, O. (2008). A central limit theorem for biased random walks on Galton–Watson trees. Prob. Theory Relat. Fields 140, 595–629.CrossRefGoogle Scholar
Pitt, J. (1974). Multiple points of transient random walks. Proc. Amer. Math. Soc. 43, 195–199.CrossRefGoogle Scholar
Sznitman, A. and Zerner, M. (1991). A law of large numbers for random walks in random environment. Ann. Prob. 27, 1851–1869.Google Scholar
Whitt, W. (2002). Stochastic-Process Limits. Springer, New York.CrossRefGoogle Scholar
Zindy, O. (2009). Scaling limit and aging for directed trap models. Markov Process. Relat. Fields 15, 31–50.Google Scholar