标题： 动态离散网络 显示英文标题

标题： The Dynamical Discrete Web

摘要： 动力离散网络（DDW），由Howitt和Warren最近的工作引入，是一组合并的简单对称一维随机游走，在一个额外的连续动态参数s中演化。演化是通过独立更新定义在离散时空上的基础伯努利变量来实现的，这些变量在任何固定的s下定义离散网络。在本文中，我们研究了异常（随机）s值的存在性，此时网络的路径不表现出通常的随机游走行为，以及此类异常s集合的豪斯多夫维数。我们的结果受到Häggstrom、Peres和Steif关于高维动态渗流的异常时间的结果以及Schramm和Steif在二维情况下的结果的启发。 DDW中游走的异常行为与Benjamini、Häggstrom、Peres和Steif的动态随机游走的情况有很大不同。 特别是，我们证明存在一些异常的s值，使得从原点出发的游走S^s(n)满足limsup S^s(n)/\sqrt n \leq K，并且豪斯多夫维数与K有非平凡的依赖关系。我们还讨论了这些和其他结果如何扩展到动力布朗网络，这是DDW的一个自然标度极限。 标度极限是正在准备的一篇论文的重点；它由Howitt和Warren研究过，并与Sun和Swart的布朗网络有关。 显示英文摘要

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 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.