Quenched scaling limits of trap models

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].