From Local Mixing to Global Mixing, Uniform Spanning Forests, Exclusion Processes and Random Connected Spanning Subgraphs

研究者将研究概率中的几个问题，其中大多数涉及马尔可夫链的混合时间。第一个问题是开发工具，利用马尔可夫链的局部混合特性来理解混合时间全局意义上的收敛性。研究者希望应用这些工具来分析自旋系统中空间混合和时间混合之间的联系。研究者还将研究与排除过程有关的问题。一个例子是奥尔德斯的猜想，即在对称不相容过程中，谱隙不依赖于粒子的数量。进一步的问题涉及不对称排斥；对于其中一些模型，混合时间被认为低于相应的对称版本，但现有技术无法验证这一点。近年来，关于马尔可夫链的混合时间的计算已经发展了大量的数学研究。说明这一研究体系的一个问题是，确定需要多少次洗牌来随机化一副牌。在这方面，数学家取得了巨大的成功，为许多洗牌模型找到了非常精确的答案。然而，对混合时间的兴趣并不局限于洗牌，因为马尔可夫链是一个在广泛领域应用的关键工具。马尔可夫链的模拟每年消耗大量的计算机周期，并应用于计算机算法、统计学和统计物理等领域。

The investigator will study several problems in probability, most of which concern mixing times for Markov chains. The first problem is to develop tools for using local mixing properties of a Markov chain to understand convergence in the global sense of the mixing time. The investigator hopes to apply these tools to analyze the connection between spatial mixing and temporal mixing in spin sytems. The investigator will also study problems relating to exclusion processes. One example is Aldous's conjecture that in the symmetric exclusion process the spectral gap does not depend on the number of particles. Further problems concern asymmetric exclusion; for some of these models the mixing time is believed to be lower than the corresponding symmetric version but existing techniques cannot verify this. In recent years a large body of mathematics has been developed relating to finding the mixing time of a Markov chain. A problem that illustrates this body of research is that of determining how many shuffles are necessary to randomize a deck of cards. Here, mathematicians have had great success, finding very precise answers for many models of card shuffling. However, interest in mixing times is not limited to card shuffling since Markov chains are a crucial tool for applications in a wide range of areas. Simulations of Markov chains consume an enormous number of computer cycles each year and are applied in such areas as computer algorithms, statistics and statistical physics.