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January 2011 Quenched scaling limits of trap models
Milton Jara, Claudio Landim, Augusto Teixeira
Ann. Probab. 39(1): 176-223 (January 2011). DOI: 10.1214/10-AOP554

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].

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Milton Jara. Claudio Landim. Augusto Teixeira. "Quenched scaling limits of trap models." Ann. Probab. 39 (1) 176 - 223, January 2011. https://doi.org/10.1214/10-AOP554

Information

Published: January 2011
First available in Project Euclid: 3 December 2010

zbMATH: 1211.60040
MathSciNet: MR2778800
Digital Object Identifier: 10.1214/10-AOP554

Subjects:
Primary: 60F99 , 60G50 , 60G52 , 60J27 , 60K35 , 60K37 , 82C05 , 82C41 , 82D30

Keywords: gap diffusions , hydrodynamic equation , metastability , Scaling limit , trap models

Rights: Copyright © 2011 Institute of Mathematical Statistics

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Vol.39 • No. 1 • January 2011
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