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 $W_x^N$, $x=(x_1,...,x_d)$, $0\le x_i<N$, 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 $(N{W}_x^N{)}^{-1}$, 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 $(W_x^N{)}^{-1}$ has a metastable behavior, as defined in \cite{bl1}, described by the $K$-process introduced in \cite{fm1}.

References in corpus (3)

• K-processes, scaling limit and aging for the trap model in the complete graph
• Hydrodynamic behavior of one dimensional subdiffusive exclusion processes with random conductances
• Hydrodynamic limit of gradient exclusion processes with conductances on $\bb Z^d$

Cited by in corpus (1)

• Principal eigenvalue for random walk among random traps on Z^d