QUENCHED SCALING LIMITS OF TRAP MODELS

• 成果类型: Article
• 署名作者: Jara, Milton; Landim, Claudio; Teixeira, Augusto
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite PSL; Universite Paris-Dauphine; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite de Rouen Normandie; Swiss Federal Institutes of Technology Domain; ETH Zurich
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/10-AOP554
• 发表日期: 2011
• 页码: 176-223
• 关键词: bouchauds model; discrete torus; random-walk; dimension; BEHAVIOR

摘要

In this paper, we study Bouchaud's trap model on the discrete d-dimensional torus T-n(d) = (Z/nZ)(d). In this process, a particle performs a symmetric simple random walk, which waits at the site x is an element of T-n(d), an exponential time with mean xi x, where {xi x, x is an element of T-n(d)} is a realization of an i.i.d. sequence of positive random variables with an alpha-stable law. Intuitively speaking, the value of xi x 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 Beltran 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].