Quenched scaling limits of trap models

Abstract

Fix a strictly positive measure $W$ on the $d$-dimensional torus $\bb T^d$. For an integer $N\ge 1$, denote by $W^N_x$, $x=(x_1, ..., x_d)$, $0\le x_i <N$, the $W$-measure of the cube $[x/N, (x+\mb 1)/N)$, where $\mb 1$ is the vector with all components equal to 1. In dimension 1, we prove that the hydrodynamic behavior of a superposition of independent random walks, in which a particle jumps from $x/N$ to one of its neighbors at rate $(N W^N_x)^{-1}$, is described in the diffusive scaling by the linear differential equation $\partial_t \rho = (d/dW)(d/dx) \rho$. In dimension $d>1$, if $W$ is a finite discrete measure, $W=\sum_{i\ge 1} w_i \delta_{x_i}$, we prove that the random walk which jumps from $x/N$ uniformly to one of its neighbors at rate $(W^N_x)^{-1}$ has a metastable behavior, as defined in \cite{bl1}, described by the $K$-process introduced in \cite{fm1}.