Stochastic processes and geometry of random networks

我的研究重点是离散和连续物体之间的界面，以及随机和有序结构的几何和拓扑性质之间的联系，以及它们上随机过程的行为。*平面地图。这里的一个主要目标是开发用于分析随机平面地图和量子重力的新工具，量子重力是一个蓬勃发展的领域，它是概率论、统计物理学、组合学和复数分析的交叉点。一个关键的想法是将平面地图视为曲面而不是度量空间，无论是通过赋予地图保角结构，还是通过使用Koebe的圆填充定理将地图嵌入R2。这导致了这一领域的中心开放问题，如随机平面地图上简单随机游动的速度指数和尺度极限的确定，以及KPZ恒等式，一个关于随机地图和Z2中某些随机集合的维度的猜想公式，其中计算维度是出了名的困难问题。*地图的尺度。*第二个目标是将最近的结果扩展到其他类型的地图，这些结果给出了随机地图的比例限制。虽然已经对一些类型的地图进行了深入的分析，但研究更一般的随机平面地图的大比例尺结构的工具仍然很少。特别是，我计划研究被赋予一些统计物理模型的地图，如生成树或独立集。*测地线网络。第三个目标是更好地理解布朗图中测地线的结构。众所周知，在布朗地图中，测地线到某一点通常在中途结合，而且几乎没有一点在任何测地线的内部。我打算扩展这一点，研究测地线网络，特别是证明所有没有端点的测地线的并具有1维。*高亏格地图。一个由来已久的猜想是，高亏格的随机映射有一个有限的分布极限，并且这个极限有一个猜想的分布。为了证明这一点，我将在与Ray和其他人在单细胞图谱上的工作的基础上再接再厉。下一步是了解从单细胞映射到Z的图的同胚图的个数。*群的随机游动和几何。可变性和Liouville性质(每个有界调和函数都是常数)，是一些群的两个基本几何性质。众所周知，很难从总体上确定一个团体是否愿意服从，以及它是否是刘维尔。一个主要目标是证明成为刘维尔并不取决于对一个群的生成集的选择。我继续研究群体上的随机行走，以及群体行动的Schreier图。通过证明Schreier图是循环的，我们可以证明某些群是Liouville的，因此也是可服从的。我将把这些方法的适用性扩展到其他不知道可改善性和可改性的群，如Thompson群，区间交换群，以及Sidki猜测的某些自动机群是可服从的。**

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.**