Exceptional Times for the Discrete Web and Predictability in Ising Models

摘要

The dynamical discrete web (DyDW) is a system of one-dimensional coalescing random walks that evolves in an extra dynamical time parameter, tau. At any deterministic tau the paths behave as coalescing simple symmetric random walks. It has been shown by Fontes, Newman, Ravishankar and Schertzer that there exist exceptional dynamical times, tau, at which the path from the origin is K-subdiffusive, meaning the path is bounded above by j plus the square root of t for all t, where t is the random walk time, and j is some constant. In this thesis we consider for the first time the existence of superdiffusive exceptional times. To be specific, we consider tau such that the limsup of the path from the origin divided by the square root of t times the log of t is greater than or equal to C. We show that such exceptional times exist for small values of C, but they do not exist for large C. Another goal of this thesis is to establish the existence of exceptional times for which the path from the origin is K-subdiffusive in both directions, i.e., tau such that absolute value of the path from the origin is bounded above by j plus the square root of t 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. This thesis concludes with a brief description of recent joint work with Charles Newman and Daniel Stein on dynamical Ising models. We consider Ising models with symmetric i.i.d. initial conditions evolving under zero temperature dynamics. The main goal is to examine the relative importance of the initial conditions versus the dynamics in determining the state of the system at large times.