Quenched scaling limits of trap models

Fix a strictly positive measure $W$ on the $d$-dimensional torus $\text{bb}{T}^d$. For an integer $N\ge 1$, denote by $W_x^N$, $x=(x_1, …, x_d)$,$0\le x_i <N$,the$W$-measureofthecube$[x/N, (x+\mb 1)/N)$,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 $x\mathrm{/}N$ to one of its neighbors at rate $(N{W}_x^N{)}^{-1}$, is described in the diffusive scaling by the linear differential equation ${\mathrm{\partial }}_t\rho =(d\mathrm{/}dW)(d\mathrm{/}dx)\rho$. In dimension $d>1$, if $W$ is a finite discrete measure, $W=\sum_{i\ge 1} w_i \delta_{x_i}$,weprovethattherandomwalkwhichjumpsfrom$x/N$ 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}.