Superdiffusive and Subdiffusive Exceptional Times in the Dynamical Discrete Web

TL;DR

This paper investigates the existence and properties of exceptional times in the dynamical discrete web where paths exhibit superdiffusive or subdiffusive behavior, extending previous work on subdiffusive times and analyzing their fractal dimensions.

Contribution

It introduces new results on superdiffusive exceptional times, provides bounds on their Hausdorff dimensions, and extends understanding of anomalous diffusive behaviors in the dynamical web.

Findings

Existence of superdiffusive exceptional times.

Bounds on Hausdorff dimensions of subdiffusive times.

Lower bounds on Hausdorff dimension of superdiffusive times.

Abstract

The dynamical discrete web is a system of one-dimensional coalescing random walks that evolves in an extra dynamical time parameter. At any deterministic dynamical time, the paths behave as coalescing simple symmetric random walks. This paper studies the existence of (random) exceptional dynamical times at which the paths violate certain almost sure properties of random walks. It was shown in 2009 by Fontes, Newman, Ravishankar and Schertzer that there exist exceptional times at which the path starting from the origin violates the law of the iterated logarithm. Their results gave exceptional times at which the path is slightly subdiffusive in one direction. This paper extends this to obtain times at which the path is slightly superdiffusive in one direction and times at which the path is slightly subdiffusive in both directions. We also obtain upper and lower bounds for the Hausdorff dimensions of the sets of two-sided subdiffusive exceptional times, and a lower bound for the Hausdorff dimension of set of superdiffusive exceptional times.