动态离散网络
The Dynamical Discrete Web
摘要：
动力离散网（DDW）是Howitt和Warren在最近的工作中提出的一个简单对称一维随机游动的聚合系统，它在一个额外的连续动力学参数s中演化。演化是通过独立更新的基础伯努利变量索引的离散空间-时间，定义在任何固定的s离散网络。在本文中，我们研究的存在例外（随机）值的网络的道路不像通常的随机游动和豪斯多夫维数的这种例外s的集合。我们的结果是由Häggstrom，Peres和Steif的高维动力学渗流的例外时间，以及Schramm和Steif的二维动力学渗流的例外时间所激发的。DDW中游动的例外行为与Benjamini、Häggstrom、Peres和Steif的动态随机游动的情况有很大不同。特别地，我们证明了存在s的例外值，使得从原点S^s（n）开始的行走有limsup S^s（n）/ \sqrt n \leq K，且豪斯多夫维数对K有非平凡的依赖性.我们还讨论了如何将这些和其他结果扩展到动态布朗网络，DDW的自然尺度限制。标度极限是一篇正在准备的论文的重点;它是由Howitt和Warren研究的，与Sun和Swart的布朗网络有关。
英文摘要：
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 Häggstrom, 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, Häggstrom, 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.
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