Stochastic processes and geometry of random networks

随机过程和随机网络的几何

• 批准号: RGPIN-2015-04570
• 负责人: Angel, Omer
• 金额: $ 1.82万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=643978
• 关键词: Stochastic; processes; geometry; random; networks

My research focuses on the interface between discrete and continuous objects, and on connections between the geometry and topological properties of structures, both random and ordered, and the behaviour of stochastic processes on them.***Planar maps.  A main aim here is the development of new tools for analysis of random planar maps and quantum gravity, a thriving field at the intersection of probability, statistical physics, combinatorics and complex analysis.  A key idea is view planar maps as a surface rather than as a metric space, whether by endowing the maps with a conformal structure, or by using Koebe's circle packing theorem to embed the maps in R2.  This leads to central open problems in this field such as determination of the speed exponent and scaling limit for the simple random walk on random planar maps, and to the KPZ identity, a conjectural formula relating dimensions of certain random sets in random maps and in Z2, where computing the dimensions are notoriously hard problems.***Scaling of maps.  A second aim is to extend recent results giving the scaling limits of random maps to other classes of maps.  While some types of maps have been analyzed, there are still very few tools for studying the large scale structure of more general random planar maps.  In particular, I plan to study maps endowed with some statistical physical model such as spanning trees or independent sets.****Geodesic networks.  A third aim is to better understand the structure of geodesics in the Brownian map.  It is known that in the Brownian map geodesics to a point typically coalesce midway, and that almost no point is in the interior of any geodesic.  I intend to extend this, and study geodesics network, and in particular show that the union of all geodesics without their endpoints has dimension 1.****High genus maps.  A long standing conjecture is that random maps of high genus have a distributional limit, and there is a conjectured distribution for the limit.  In order to prove this I will build on work with Ray and others on unicellular maps.  The next step is to understand the number of graph homeomorphisms from the unicellular maps to Z.****Random walks and geometry of groups.  Amenability and the Liouville property (every bounded harmonic function is constant), are two fundamental geometric properties of some groups.  It is notoriously hard to determine in general whether a group is amenable and if it is Liouville.  A major goal is to prove that being Liouville does not depend on the choice of a generating set of a group.  I proceed by studying random walks on the groups, and on Schreier graphs for actions of the groups.  By showing that the Schreier graphs are recurrent we can prove that certain groups are Liouville, and are thus also amenable.  I will expand the applicability of these methods to other groups for which amenability and Liouville are not known, such as Thompson's group, and the interval exchange group, and certain automaton groups conjectured by Sidki to be amenable.**

我的研究重点是离散和连续对象之间的接口，以及结构的几何和拓扑性质之间的连接，随机和有序，以及随机过程对它们的行为。平面地图。 这里的一个主要目标是开发用于分析随机平面映射和量子引力的新工具，量子引力是概率、统计物理、组合学和复分析交叉的一个蓬勃发展的领域。 一个关键的想法是将平面映射视为曲面而不是度量空间，无论是通过赋予映射共形结构，还是通过使用Koebe的圆填充定理将映射嵌入R2中。 这导致了该领域的中心开放问题，例如确定随机平面映射上简单随机游动的速度指数和标度极限，以及KPZ恒等式，一个与随机映射和Z2中某些随机集的维数有关的数学公式，其中计算维数是众所周知的困难问题。地图的比例。 第二个目的是扩展最近的结果，使缩放限制的随机映射到其他类别的地图。 虽然已经分析了某些类型的地图，但仍然有很少的工具来研究更一般的随机平面地图的大尺度结构。 特别是，我计划研究具有某些统计物理模型（例如生成树或独立集）的地图。*测地线网络。 第三个目的是更好地理解布朗映射中测地线的结构。 众所周知，在布朗映射中，到一点的测地线通常在中途合并，并且几乎没有点在任何测地线的内部。 我打算扩展这一点，并研究测地线网络，特别是表明，所有测地线的联合没有端点有维度1。高属映射。 一个长期存在的猜想是高亏格的随机映射有一个分布极限，并且这个极限有一个严格的分布。 为了证明这一点，我将与雷和其他人一起在单细胞地图上工作。 下一步是了解从单胞映射到Z的图同胚的数量。群的随机游动和几何。 顺从性和Liouville性质（每个有界调和函数都是常数）是某些群的两个基本几何性质。 众所周知，一般来说，很难确定一个群体是否顺从，如果它是刘维。 一个主要的目标是证明刘维并不依赖于一个组的生成集的选择。 我继续研究随机游走的群体，并对施赖尔图的行动的群体。 通过证明Schreier图是递归的，我们可以证明某些群是刘维尔群，因而也是顺从的。 我将把这些方法的适用性扩展到其他不知道顺从性和刘维的群，如汤普森群、区间交换群和某些由西德基证明为顺从的自动机群。