The Dynamical Discrete Web

TL;DR

This paper investigates the existence and properties of exceptional times in the dynamical discrete web where the paths deviate from typical behavior, analyzing their Hausdorff dimension and implications for the dynamical Brownian web.

Contribution

It establishes the existence of exceptional times with nontrivial Hausdorff dimension in the dynamical discrete web and explores their properties and extensions to the dynamical Brownian web.

Findings

Existence of exceptional s where paths deviate from typical behavior.

Hausdorff dimension of the set of exceptional s depends on the deviation parameter.

Extension of results to the dynamical Brownian web and its scaling limit.

Abstract

The dynamical discrete web (DDW), introduced in recent work of Howitt and Warren, is a system of coalescing simple symmetric one-dimensional random walks which evolve in an extra continuous dynamical parameter s. The evolution is by independent updating of the underlying Bernoulli variables indexed by discrete space-time that define the discrete web at any fixed s. In this paper, we study the existence of exceptional (random) values of s where the paths of the web do not behave like usual random walks and the Hausdorff dimension of the set of such exceptional s. Our results are motivated by those about exceptional times for dynamical percolation in high dimension by H\"aggstrom, Peres and Steif, and in dimension two by Schramm and Steif. The exceptional behavior of the walks in DDW is rather different from the situation for dynamical random walks of Benjamini, H\"aggstrom, Peres and Steif. In particular, we prove that there are exceptional values of s for which the walk from the origin S^s(n) has limsup S^s(n)/\sqrt n \leq K with a nontrivial dependence of the Hausdorff dimension on K. We also discuss how these and other results extend to the dynamical Brownian web, a natural scaling limit of DDW. The scaling limit is the focus of a paper in preparation; it was studied by Howitt and Warren and is related to the Brownian net of Sun and Swart.