Universality of trap models in the ergodic time scale

Abstract

Consider a sequence of possibly random graphs $G_{N}=(V_{N},E_{N})$, $N\ge1$, whose vertices’s have i.i.d. weights $\{W^{N}_{x}:x\in V_{N}\}$ with a distribution belonging to the basin of attraction of an $\alpha$-stable law, $0<\alpha<1$. Let $X^{N}_{t}$, $t\ge0$, be a continuous time simple random walk on $G_{N}$ which waits a mean $W^{N}_{x}$ exponential time at each vertex $x$. Under considerably general hypotheses, we prove that in the ergodic time scale this trap model converges in an appropriate topology to a $K$-process. We apply this result to a class of graphs which includes the hypercube, the $d$-dimensional torus, $d\ge2$, random $d$-regular graphs and the largest component of super-critical Erdős–Rényi random graphs.