Quenched scaling limits of trap models

TL;DR

This paper studies the scaling limits of trap models on the torus, showing diffusive behavior in one dimension and metastability in higher dimensions with discrete traps.

Contribution

It proves hydrodynamic limits for trap models in one dimension and characterizes metastable behavior in higher dimensions with discrete traps.

Findings

Hydrodynamic limit in 1D described by a differential equation.

Metastable behavior in higher dimensions with discrete traps.

Connection to K-processes for describing metastability.

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}.