### Abstract

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 S_{0}
^{τ} 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.

Original language | English (US) |
---|---|

Pages (from-to) | 2832-2858 |

Number of pages | 27 |

Journal | Stochastic Processes and their Applications |

Volume | 119 |

Issue number | 9 |

DOIs | |

State | Published - Sep 2009 |

### Fingerprint

### Keywords

- Brownian web
- Coalescing random walks
- Dynamical random walks
- Exceptional times
- Hausdorff dimension
- Law of the iterated logarithm
- Sticky random walks

### ASJC Scopus subject areas

- Modeling and Simulation
- Statistics and Probability
- Applied Mathematics

### Cite this

*Stochastic Processes and their Applications*,

*119*(9), 2832-2858. https://doi.org/10.1016/j.spa.2009.03.001

**Exceptional times for the dynamical discrete web.** / Fontes, L. R G; Newman, Charles; Ravishankar, K.; Schertzer, E.

Research output: Contribution to journal › Article

*Stochastic Processes and their Applications*, vol. 119, no. 9, pp. 2832-2858. https://doi.org/10.1016/j.spa.2009.03.001

}

TY - JOUR

T1 - Exceptional times for the dynamical discrete web

AU - Fontes, L. R G

AU - Newman, Charles

AU - Ravishankar, K.

AU - Schertzer, E.

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 S0 τ 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 S0 τ 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

UR - http://www.scopus.com/inward/record.url?scp=68349129917&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=68349129917&partnerID=8YFLogxK

U2 - 10.1016/j.spa.2009.03.001

DO - 10.1016/j.spa.2009.03.001

M3 - Article

VL - 119

SP - 2832

EP - 2858

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

IS - 9

ER -