TY - JOUR

T1 - Exceptional times for the dynamical discrete web

AU - Fontes, L. R.G.

AU - Newman, C. M.

AU - Ravishankar, K.

AU - Schertzer, E.

N1 - Funding Information:
The research of L.R.G. Fontes was supported in part by FAPESP grant 2004/07276-2 and CNPq grants 307978/2004-4 and 484351/2006-0. The research of the other authors was supported in part by N.S.F. grants DMS-01-04278 and DMS-06-06696. The authors thank a referee and associate editor for useful comments.

PY - 2009/9

Y1 - 2009/9

N2 - The dynamical discrete web (DyDW), introduced in the recent work of Howitt and Warren, is a system of coalescing simple symmetric one-dimensional random walks which evolve in an extra continuous dynamical time parameter τ. The evolution is by independent updating of the underlying Bernoulli variables indexed by discrete space-time that define the discrete web at any fixed τ. In this paper, we study the existence of exceptional (random) values of τ where the paths of the web do not behave like usual random walks and the Hausdorff dimension of the set of such exceptional τ. Our results are motivated by those about exceptional times for dynamical percolation in high dimension by Häggstrom, Peres and Steif, and in dimension two by Schramm and Steif. The exceptional behavior of the walks in the DyDW is rather different from the situation for the dynamical random walks of Benjamini, Häggstrom, Peres and Steif. For example, we prove that the walk from the origin S0τ violates the law of the iterated logarithm (LIL) on a set of τ of Hausdorff dimension one. We also discuss how these and other results should extend to the dynamical Brownian web, the natural scaling limit of the DyDW.

AB - The dynamical discrete web (DyDW), introduced in the recent work of Howitt and Warren, is a system of coalescing simple symmetric one-dimensional random walks which evolve in an extra continuous dynamical time parameter τ. The evolution is by independent updating of the underlying Bernoulli variables indexed by discrete space-time that define the discrete web at any fixed τ. In this paper, we study the existence of exceptional (random) values of τ where the paths of the web do not behave like usual random walks and the Hausdorff dimension of the set of such exceptional τ. Our results are motivated by those about exceptional times for dynamical percolation in high dimension by Häggstrom, Peres and Steif, and in dimension two by Schramm and Steif. The exceptional behavior of the walks in the DyDW is rather different from the situation for the dynamical random walks of Benjamini, Häggstrom, Peres and Steif. For example, we prove that the walk from the origin S0τ violates the law of the iterated logarithm (LIL) on a set of τ of Hausdorff dimension one. We also discuss how these and other results should extend to the dynamical Brownian web, the natural scaling limit of the DyDW.

KW - Brownian web

KW - Coalescing random walks

KW - Dynamical random walks

KW - Exceptional times

KW - Hausdorff dimension

KW - Law of the iterated logarithm

KW - Sticky random walks

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U2 - 10.1016/j.spa.2009.03.001

DO - 10.1016/j.spa.2009.03.001

M3 - Article

AN - SCOPUS:68349129917

VL - 119

SP - 2832

EP - 2858

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

IS - 9

ER -