The Annals of Probability

Quenched scaling limits of trap models

Milton Jara, Claudio Landim, and Augusto Teixeira

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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].

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

First available in Project Euclid: 3 December 2010

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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]

Trap models scaling limit hydrodynamic equation gap diffusions metastability


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.

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