Geometry of Random Curves

提案：DMS-9971493主要研究者：Almut Burchard摘要：从（连续）尺度限制的角度考虑在晶格上构造的随机曲线，其中晶格间距取为零。 第一个问题是刻画相当一般的随机曲线系统的标度极限。 问题的一个方面是确定一个给定的（不一定是随机的）曲线的规律性和尺寸，而实际上并没有参数化它。此外，一系列的标度指数，这是相关的规律性和紧密性将被研究。第二个问题是特别关注的随机生成树在两个维度。在许多例子中，存在关于指数值的精确计算。其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，两个标准例子，称为最小（随机）生成树和均匀随机生成树，在极限上是不同还是一致。此外，现有的方法应该扩展到覆盖一类随机图上的树过程，具有自然的共形不变性。关于这些问题的工作将是持续合作的一部分。 第三个问题是试图建立任何非平凡的约束的自避免随机游走的维度，适应技术是成功的随机生成树和循环擦除行走。最后，研究人员将参与旨在了解高维临界渗流中的连接曲线的合作努力。随机曲线提供了分形集的一些最自然的例子。它们出现在随机生成树中（每对点都由唯一的曲线连接），临界渗流簇中（如果两个点通过沿着被占用的键的路径连接，则它们在同一簇中），作为随机行走，以及作为平面的随机子集之间的界面（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构造在有限大小的盒子内的精细晶格上。模拟表明，在许多感兴趣的模型，一个有意义的连续极限出现的晶格间距为零。人们普遍认为，这些极限具有显著的对称性，并且它们应该允许简单的描述。然而，只有部分结果保证这种极限存在，并且关于它们的性质的许多问题仍然是开放的。

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.