The Annals of Probability

Quenched scaling limits of trap models

Milton Jara, Claudio Landim, and Augusto Teixeira

Full-text: Open access

Abstract

In this paper, we study Bouchaud’s trap model on the discrete d-dimensional torus ${\mathbb{T}}^{d}_{n}=({\mathbb{Z}}/n{\mathbb{Z}})^{d}$. In this process, a particle performs a symmetric simple random walk, which waits at the site $x\in {\mathbb{T}}^{d}_{n}$ an exponential time with mean ξx, where $\{\xi_{x},x\in {\mathbb{T}}^{d}_{n}\}$ is a realization of an i.i.d. sequence of positive random variables with an α-stable law. Intuitively speaking, the value of ξx gives the depth of the trap at x. In dimension d=1, we prove that a system of independent particles with the dynamics described above has a hydrodynamic limit, which is given by the degenerate diffusion equation introduced in [Ann. Probab. 30 (2002) 579–604]. In dimensions d>1, we prove that the evolution of a single particle is metastable in the sense of Beltrán and Landim [Tunneling and Metastability of continuous time Markov chains (2009) Preprint]. Moreover, we prove that in the ergodic scaling, the limiting process is given by the K-process, introduced by Fontes and Mathieu in [Ann. Probab. 36 (2008) 1322–1358].

Article information

Source
Ann. Probab., Volume 39, Number 1 (2011), 176-223.

Dates
First available in Project Euclid: 3 December 2010

Permanent link to this document
https://projecteuclid.org/euclid.aop/1291388300

Digital Object Identifier
doi:10.1214/10-AOP554

Mathematical Reviews number (MathSciNet)
MR2778800

Zentralblatt MATH identifier
1211.60040

Subjects
Primary: 60F99: None of the above, but in this section 60G50: Sums of independent random variables; random walks 60G52: Stable processes 60J27: Continuous-time Markov processes on discrete state spaces 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K37: Processes in random environments 82C05: Classical dynamic and nonequilibrium statistical mechanics (general) 82D30: Random media, disordered materials (including liquid crystals and spin glasses) 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

Keywords
Trap models scaling limit hydrodynamic equation gap diffusions metastability

Citation

Jara, Milton; Landim, Claudio; Teixeira, Augusto. Quenched scaling limits of trap models. Ann. Probab. 39 (2011), no. 1, 176--223. doi:10.1214/10-AOP554. https://projecteuclid.org/euclid.aop/1291388300


Export citation

References

  • [1] Aldous, D. J. and Fill, J. Reversible Markov Chains and Random Walks on Graphs. Available at http://www.stat.berkeley.edu/users/aldous/RWG/book.html.
  • [2] Beltrán, J. and Landim, C. (2010). Tunneling and Metastability of continuous time Markov chains. J. Stat. Phys. 140 1065–1114.
  • [3] Ben Arous, G. and Černý, J. (2005). Bouchaud’s model exhibits two different aging regimes in dimension one. Ann. Appl. Probab. 15 1161–1192.
  • [4] Ben Arous, G. and Černý, J. (2006). Dynamics of trap models. In Mathematical Statistical Physics 331–394. Elsevier, Amsterdam.
  • [5] Ben Arous, G. and Černý, J. (2007). Scaling limit for trap models on ℤd. Ann. Probab. 35 2356–2384.
  • [6] 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.
  • [7] Ben Arous, G., Černý, J. and Mountford, T. (2006). Aging in two-dimensional Bouchaud’s model. Probab. Theory Related Fields 134 1–43.
  • [8] Benjamini, I. and Sznitman, A.-S. (2008). Giant component and vacant set for random walk on a discrete torus. J. Eur. Math. Soc. (JEMS) 10 133–172.
  • [9] Černý, J. (2006). The behaviour of aging functions in one-dimensional Bouchaud’s trap model. Comm. Math. Phys. 261 195–224.
  • [10] Faggionato, A., Jara, M. and Landim, C. (2009). Hydrodynamic behavior of one dimensional subdiffusive exclusion processes with random conductances. Probab. Theory Related Fields. 144 633–667.
  • [11] 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.
  • [12] Fontes, L. R. G. and Lima, P. H. S. (2009). Convergence of symmetric trap models in the hypercube. In New Trends in Mathematical Physics (V. Sidoravičius, ed.) 285–297. Springer, Netherlands.
  • [13] Fontes, L. R. G. and Mathieu, P. (2008). K-processes, scaling limit and aging for the trap model in the complete graph. Ann. Probab. 36 1322–1358.
  • [14] Franco, T. and Landim, C. (2010). Hydrodynamic limit of gradient exclusion processes with conductances. Arch. Ration. Mech. Anal. 195 409–439.
  • [15] Kipnis, C. and Landim, C. (1999). Scaling Limits of Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 320. Springer, Berlin.
  • [16] Lawler, G. F. (1991). Intersections of Random Walks. Birkhäuser, Boston, MA.
  • [17] Levin, D. A., Peres, Y. and Wilmer, E. L. (2009). Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI.
  • [18] Mourrat, J. C. (2009). Principal eigenvalue for random walk among random traps on ℤd. Potential Anal. To appear. Available at arXiv:0805.0706v1.
  • [19] Papanicolaou, G. C. and Varadhan, S. R. S. (1982). Diffusions with random coefficients. In Statistics and Probability: Essays in Honor of C. R. Rao 547–552. North-Holland, Amsterdam.
  • [20] Stone, C. (1963). Limit theorems for random walks, birth and death processes, and diffusion processes. Illinois J. Math. 7 638–660.
  • [21] Valentim, F. (2011). Hydrodynamic limit of gradient exclusion processes with conductances on ℤd. Ann. Inst. H. Poincoré Probab. Statist. To appear.
  • [22] Vázquez, J. L. (2007). The Porous Medium Equation: Mathematical Theory. Oxford Univ. Press, Oxford.
  • [23] Windisch, D. (2008). Random walk on a discrete torus and random interlacements. Electron. Commun. Probab. 13 140–150.