  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.
