Universality of trap models in the ergodic time scale (Q471153)

Let \(\{G_N:N\geq 1\}\), \(G_N=(V_N,E_N)\), be a sequence of random, finite, connected graphs, where \(V_N\) is the set of vertices, \(E_N\) the set of unoriented edges. The number \(|V_N|\) of vertices converges to \(+\infty \) in probability. The vertices have weights \(\{W_x^N:x\in V_N\}\) with a distribution belonging to the basin of attraction of an \(\alpha \)-stable law. For each \(N\), there is a continuous-time random walk on \(V_N\) which waits a mean \(W_x^N\) at \(x\), then it jumps to one of its neighbors with uniform probability. Such an environment can be interpreted as a landscape of valleys and traps whose depths are given by the random variables \(W_x\). The main result of the paper is that the random walk under rather general conditions converges to a \(K\)-process. Sufficient conditions are given for the convergence to the \(K\)-process for the sequence of random walks among random traps. This result is adapted to random pseudo-transitive graphs and graphs with asymptotically random conductances. This includes the largest component of super-critical Erdős-Rényi graphs.