Geometry of Random Curves

摘要：在晶格上构造的随机曲线将从晶格间距取为零的（连续体）缩放极限的角度来考虑。第一个问题是描述相当一般的随机曲线系统的标度极限。该问题的一个方面是在没有实际参数化的情况下确定给定（不一定是随机的）曲线的规律性和维度。此外，还研究了与正则性和紧密性相关的标度指数序列。第二个问题是关于二维的随机生成树。在许多例子中，存在着关于指数值的精确推测。目的是证实或反驳这些预测，特别是前三个非平凡指数。一个相关的问题是，两个标准示例（称为最小（随机）生成树和均匀随机生成树）的缩放限制是否在极限上不同或重合。此外，现有的方法应该扩展到一类具有自然保形不变性的随机图上的树过程。关于这些问题的工作将是持续合作的一部分。第三个问题是尝试在自避免随机行走的维度上建立任何非平凡的边界，采用在随机生成树和循环擦除行走中成功的技术。最后，研究者将参与一项旨在理解高维临界渗流中的连接曲线的合作努力。随机曲线提供了一些最自然的分形集的例子。它们出现在随机生成树中（其中每对点都由一条唯一的曲线连接），在临界渗透簇中（其中两个点在同一簇中，如果它们由沿已占据键的路径连接），作为随机行走，以及作为平面随机子集之间的接口（例如布朗运动的边界，或随机函数的水平集）。这些随机曲线系统通常是在有限大小的盒子内的精细晶格上构造的。模拟表明，在许多感兴趣的模型中，当晶格间距为零时，会出现一个有意义的连续体极限。人们普遍认为，这些极限具有显著的对称性，可以用简单的描述。然而，只有部分结果保证这样的极限存在，而且关于它们的性质的许多问题仍未解决。

Proposal: DMS-9971493Principal Investigator: Almut BurchardAbstract: Random curves constructed on a lattice will be considered from the perspective of the (continuum) scaling limit where the lattice spacing is taken to zero. The first problem is to characterize scaling limits of rather general systems of random curves. One aspect of the problem is to determine the regularity and dimension of a given (not necessarily random) curve without actually parametrizing it. Moreover, a sequence of scaling exponents which are relevant for regularity and tightness will be studied. The second problem is concerned specifically with random spanning trees in two dimensions. Precise conjectures exist regarding the values of the exponents in a number of examples. The object is to confirm or contradict those predictions, especially for the first three non-trivial exponents. A related question is whether the scaling limits of two standard examples, called the Minimal (Random) Spanning Tree, and the Uniformly Random Spanning Tree, differ or coincide in the limit. Moreover, existing methods should be extended to cover a class of tree processes on random graphs which has natural conformal invariance properties. Work on these questions will be part of a continuing collaboration. The third problem is to try to establish any non-trivial bound on the dimension of self-avoiding random walks, adapting techniques that were successful for random spanning trees and loop-erased walks. Finally, the investigator will participate in a collaborative effort aimed at understanding the connecting curves in high-dimensional critical percolation. Random curves provide some of the most natural examples of fractal sets. They appear in random spanning trees (where every pair of points is connected by a unique curve), in critical percolation clusters (where two points are in the same cluster, if they are joined by a path along occupied bonds), as random walks, and as interfaces between random subsets of the plane (such as the Frontier of Brownian Motion, or level sets of random functions). These systems of random curves are typically constructed on a fine lattice within some box of finite size. Simulations suggest that in many models of interest, a meaningful continuum limit emerges as the lattice spacing is taken to zero. It is widely believed that these limits have remarkable symmetry properties, and that they should admit simple descriptions. However, there are only partial results guaranteeing that such limits even exist, and many questions about their properties remain open.