Scaling limit for trap models on Zd\mathbb{Z}^d

We give the ``quenched'' scaling limit of Bouchaud's trap model in ${d\ge 2}$. This scaling limit is the fractional-kinetics process, that is the time change of a $d$-dimensional Brownian motion by the inverse of an independent $\alpha$-stable subordinator.Comment: Published in at http://dx.doi.org/10.1214/009117907000000024 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Critical window for the vacant set left by random walk on the configuration model

We study the simple random walk on the configuration model with given degree sequence $(d_1^n, \dots ,d_n^n)$ and investigate the connected components of its vacant set at level $u>0$. We show that the size of the maximal connected component exhibits a phase transition at level $u^*$ which can be related with the critical parameter of random interlacements on a certain Galton-Watson tree. We further show that there is a critical window of size $n^{-1/3}$ around $u^*$ in which the largest connected components of the vacant set have a metric space scaling limit resembling the one of the critical Erdős-Rényi random graph

Convergence to fractional kinetics for random walks associated with unbounded conductances

We consider a random walk among unbounded random conductances whose distribution has infinite expectation and polynomial tail. We prove that the scaling limit of this process is a Fractional-Kinetics process—that is the time change of a d-dimensional Brownian motion by the inverse of an independent α-stable subordinator. We further show that the same process appears in the scaling limit of the non-symmetric Bouchaud's trap mode