Invariant point processes, fair allocations, random-turn games and applications

这个建议描述了概率的几个领域的问题，包括随机几何，马尔可夫链和随机游戏。行列式点过程自然出现在几个设置，包括随机矩阵的特征值。 该提案的一个目标是开发鲁棒的方法来识别不需要计算所有阶的相关函数的行列式过程。 比较了两种对点过程的点分配等体积的方法，一种是基于Gale-Shapley稳定匹配算法，另一种是基于引力势。在马尔可夫链混合领域，“截止现象”，即距离stationary迅速减少，从接近1到接近零，被认为是举行在大多数自然链，但建立了极少数。 最近，随机游动和调和函数之间的经典联系已经被扩展到非线性势理论，利用博弈论和最优控制的思想。该提案的一个主要目的是发展和研究随机博弈，其价值函数，在thescaling限制，解决其他重要的PDE。随机散射（点过程），表现出clumping有有效的模型在统计，但随机散射，表现出排斥是不太好理解。 拟议的项目旨在更好地了解这些过程，以便在统计建模中更广泛地使用它们。给定一个随机散布点，我们将研究为散布点分配相等面积的不同算法。 马尔可夫链是模拟和随机算法中广泛使用的工具. 我们将研究这些链收敛到平衡点的基本特征。这些都应该对算法的分析产生影响。该项目有一个实质性的教育组成部分，在指导研究生和准备教育材料在本科水平上的马尔可夫链和博弈论。

This proposal describes problems in several areas of probability,involving stochastic geometry, Markov chains, and stochastic games.Determinantal point processes arise naturally in several settings,including eigenvalues of random matrices. One goal of the proposal isto develop robust methods to recognize determinantal processes that donot require calculating correlation functions of all orders. Twomethods of allocating equal volumes to the points of a point process,one based on the Gale-Shapley stable matching algorithm, the other ongravitational potentials, will be compared. In the area of Markovchain mixing, the ``cutoff phenomenon'' where distance to stationaritydecreases rapidly from near 1 to near zero is believed to hold in mostnatural chains but is established for very few. A robust and easilycheckable possible criterion for cutoff will be investigated.Recently, the classical connection between random walks and harmonicfunctions has been extended to nonlinear potential theory using ideasfrom game theory and optimal control. A major aim of the proposal is todevelop and study stochastic games whose value functions, in thescaling limit, solve other important PDE.Random scatters (Point processes) that exhibit clumping have effectivemodels in statistics, but random scatters that exhibit repulsion areless well understood. The proposed project aims to develop a betterunderstanding of such processes which should enable wider use for themin statistical modeling. Given a random scatter, different algorithmsfor allocating equal areas to the points of the scatter will beinvestigated. Markov chains are a widely used tool in simulation andrandomized algorithms. Basic features of convergence to equilibriumin these chains will be studied. These should have an impact on theanalysis of algorithms. The project has a substantial educationalcomponent, in mentoring of graduate students and preparation ofeducational materials at the undergraduate level on Markov chains andgame theory.