Spread of wave packets in disordered hierarchical lattices

We consider the spreading of the wave packet in the generalized Rosenzweig-Porter random matrix ensemble in the region of non-ergodic extended states $1<\gamma<2$. We show that despite non-trivial fractal dimensions $0 < D_{q}=2-\gamma<1$ characterize wave function statistics in this region, the wave packet spreading $\langle r^{2} \rangle \propto t^{\beta}$ is governed by the "diffusion" exponent $\beta=1$ outside the ballistic regime $t>\tau\sim 1$ and $\langle r^{2}\rangle \propto t^{2}$ in the ballistic regime for $t<\tau\sim 1$. This demonstrates that the multifractality exhibits itself only in {\it local} quantities like the wave packet survival probability but not in the large-distance spreading of the wave packet.Comment: Accepted in EP

The diameter of weighted random graphs

In this paper we study the impact of random exponential edge weights on the distances in a random graph and, in particular, on its diameter. Our main result consists of a precise asymptotic expression for the maximal weight of the shortest weight paths between all vertices (the weighted diameter) of sparse random graphs, when the edge weights are i.i.d. exponential random variables.Comment: Published at http://dx.doi.org/10.1214/14-AAP1034 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org