The Annals of Probability

Randomly trapped random walks

Gérard Ben Arous, Manuel Cabezas, Jiří Černý, and Roman Royfman

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Abstract

We introduce a general model of trapping for random walks on graphs. We give the possible scaling limits of these Randomly Trapped Random Walks on $\mathbb{Z}$. These scaling limits include the well-known fractional kinetics process, the Fontes–Isopi–Newman singular diffusion as well as a new broad class we call spatially subordinated Brownian motions. We give sufficient conditions for convergence and illustrate these on two important examples.

Article information

Source
Ann. Probab., Volume 43, Number 5 (2015), 2405-2457.

Dates
Received: March 2013
Revised: February 2014
First available in Project Euclid: 9 September 2015

Permanent link to this document
https://projecteuclid.org/euclid.aop/1441792289

Digital Object Identifier
doi:10.1214/14-AOP939

Mathematical Reviews number (MathSciNet)
MR3395465

Zentralblatt MATH identifier
1329.60354

Subjects
Primary: 60K37: Processes in random environments 60G52: Stable processes
Secondary: 60F17: Functional limit theorems; invariance principles

Keywords
Bouchaud trap model random walk scaling limit percolation

Citation

Ben Arous, Gérard; Cabezas, Manuel; Černý, Jiří; Royfman, Roman. Randomly trapped random walks. Ann. Probab. 43 (2015), no. 5, 2405--2457. doi:10.1214/14-AOP939. https://projecteuclid.org/euclid.aop/1441792289


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References

  • [1] Aldous, D. (1993). The continuum random tree. III. Ann. Probab. 21 248–289.
  • [2] Angel, O., Goodman, J., den Hollander, F. and Slade, G. (2008). Invasion percolation on regular trees. Ann. Probab. 36 420–466.
  • [3] Barlow, M. T. and Kumagai, T. (2006). Random walk on the incipient infinite cluster on trees. Illinois J. Math. 50 33–65 (electronic).
  • [4] Ben Arous, G. and Cabezas, M. (2014). Scaling limits for the random walks on the incipient infinite cluster and invasion percolation cluster on regular trees. Preprint.
  • [5] Ben Arous, G. and Černý, J. (2005). Bouchaud’s model exhibits two different aging regimes in dimension one. Ann. Appl. Probab. 15 1161–1192.
  • [6] Ben Arous, G. and Černý, J. (2006). Dynamics of trap models. In Mathematical Statistical Physics 331–394. Elsevier, Amsterdam.
  • [7] Ben Arous, G. and Černý, J. (2007). Scaling limit for trap models on $\mathbb{Z}^{d}$. Ann. Probab. 35 2356–2384.
  • [8] Ben Arous, G. and Černý, J. (2008). The arcsine law as a universal aging scheme for trap models. Comm. Pure Appl. Math. 61 289–329.
  • [9] Ben Arous, G., Černý, J. and Mountford, T. (2006). Aging in two-dimensional Bouchaud’s model. Probab. Theory Related Fields 134 1–43.
  • [10] Borodin, A. N. (1987). A weak invariance principle for local times. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 158 14–31, 169.
  • [11] Bouchaud, J.-P. (1992). Weak ergodicity breaking and aging in disordered systems. J. Phys. I (France) 2 1705–1713.
  • [12] Bouchaud, J.-P., Cugliandolo, L., Kurchan, J. and Mezard, M. (1998). Out of Equilibrium Dynamics in Spin-Glasses and Other Glassy Systems. World Scientific, Singapore.
  • [13] Bouchaud, J.-P. and Dean, D. S. (1995). Aging on Parisi’s tree. J. Phys. I (France) 5 265.
  • [14] Bovier, A. and Faggionato, A. (2005). Spectral characterization of aging: The REM-like trap model. Ann. Appl. Probab. 15 1997–2037.
  • [15] Černý, J. (2006). The behaviour of aging functions in one-dimensional Bouchaud’s trap model. Comm. Math. Phys. 261 195–224.
  • [16] Croydon, D. (2008). Convergence of simple random walks on random discrete trees to Brownian motion on the continuum random tree. Ann. Inst. Henri Poincaré Probab. Stat. 44 987–1019.
  • [17] Fontes, L. R. G., Isopi, M. and Newman, C. M. (2002). Random walks with strongly inhomogeneous rates and singular diffusions: Convergence, localization and aging in one dimension. Ann. Probab. 30 579–604.
  • [18] Georgii, H.-O. (1988). Gibbs Measures and Phase Transitions. de Gruyter Studies in Mathematics 9. de Gruyter, Berlin.
  • [19] Gnedenko, B. V. and Kolmogorov, A. N. (1968). Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Reading, MA.
  • [20] Jara, M., Landim, C. and Teixeira, A. (2011). Quenched scaling limits of trap models. Ann. Probab. 39 176–223.
  • [21] Kallenberg, O. (1983). Random Measures, 3rd ed. Akademie-Verlag, Berlin.
  • [22] Kallenberg, O. (1990). Exchangeable random measures in the plane. J. Theoret. Probab. 3 81–136.
  • [23] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
  • [24] Kallenberg, O. (2005). Probabilistic Symmetries and Invariance Principles. Springer, New York.
  • [25] Kesten, H. (1986). Subdiffusive behavior of random walk on a random cluster. Ann. Inst. Henri Poincaré Probab. Stat. 22 425–487.
  • [26] Meerschaert, M. M. and Scheffler, H.-P. (2004). Limit theorems for continuous-time random walks with infinite mean waiting times. J. Appl. Probab. 41 623–638.
  • [27] Montroll, E. W. and Weiss, G. H. (1965). Random walks on lattices. II. J. Math. Phys. 6 167–181.
  • [28] Mourrat, J.-C. (2011). Scaling limit of the random walk among random traps on $\mathbb{Z}^{d}$. Ann. Inst. Henri Poincaré Probab. Stat. 47 813–849.
  • [29] Stone, C. (1963). Limit theorems for random walks, birth and death processes, and diffusion processes. Illinois J. Math. 7 638–660.
  • [30] Whitt, W. (2002). Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer, New York.