Superdiffusive and subdiffusive exceptional times in the dynamical discrete web

The dynamical discrete web (DyDW) is a system of one-dimensional coalescing random walks that evolves in an extra dynamical time parameter, τ. At any deterministic τ the paths behave as coalescing simple symmetric random walks. It has been shown in Fontes et al. (2009) that there exist exceptional dynamical times, τ, at which the path from the origin, S0τ, is K-subdiffusive, meaning S0τ(t)≤j+Kt for all t, where t is the random walk time, and j is some constant. In this paper we consider for the first time the existence of superdiffusive exceptional times. To be specific, we consider τ such that lim supt→∞S0τ(t)/tlog(t)≥C. We show that such exceptional times exist for small values of C, but they do not exist for large C. The other goal of this paper is to establish the existence of exceptional times for which the path from the origin is K-subdiffusive in both directions, i.e. τ such that |S0τ(t)|≤j+Kt for all t. We also obtain upper and lower bounds for the Hausdorff dimensions of these two-sided subdiffusive exceptional times. For the superdiffusive exceptional times we are able to get a lower bound on Hausdorff dimension but not an upper bound.