Probabilistic Methods in Discrete Structures and Applications

这是一个在概率论、解析数论以及这两个领域之间的联系中发展定量结果的项目。 概率推理在数论的几个最新突破中发挥了重要作用，包括自1950年代以来一直开放的保罗·埃尔德什覆盖系统的最小模问题的解决方案。 本计画的一个方面是将分析加以改进，以研究相关的奇复盖问题。 分析数论中使用的技术可以用来研究由马尔可夫链控制的统计物理模型。 这些技术通常会对计算机模型中观察到的现象给出严格的证明，有时严格的分析会发现模型预测中的错误，这些错误是由有限的模型大小造成的。 该项目的目的是扩展最近的强估计的频谱扩展的形状的三次字段类似的频谱估计的四次和五次field.The最小模问题的解决方案相结合的矩的方法与技术，从概率组合，包括洛瓦兹本地引理和使用伪随机措施。 在开发覆盖系统的定量结果时，本项目将应用目前在加法组合学中使用的结构分解技术。 振动积分、稳相法和货车德尔科普特法在解析数论中经常被用来获得精确的估计。 这些技术往往也是有用的混合时间的马尔可夫链的渐近分析，并将继续工作的分析二维阿贝尔沙堆模型和一个新的局部极限定理的幂零群。

This is a project developing quantitative results in probability theory, analytic number theory, and connections between the two fields. Probabilistic reasoning has played an important role in several recent breakthroughs in number theory, including the solution of Paul Erdös' minimum modulus problem for covering systems, which had been open since the 1950s. One aspect of this project refines the analysis to study the related odd covering problem. Techniques used in analytic number theory can be employed to study models in statistical physics that are governed by Markov chains. Often these techniques give rigorous proofs of phenomena observed in computer models, and sometimes the rigorous analysis uncovers errors from model predictions that result from a limited model size. The project aims to extend recent strong estimates on the spectral expansion of the shape of cubic fields to similar spectral estimates for the shapes of quartic and quintic fields.The solution of the minimum modulus problem combines the method of moments with techniques from probabilistic combinatorics, including the Lovasz local lemma and the use of pseudo-random measures. In developing quantitative results regarding covering systems, this project will apply structural decomposition techniques currently in use in additive combinatorics. Oscillatory integrals, the method of stationary phase, and van der Corput's method are used frequently in analytic number theory to obtain sharp estimates. These techniques are often useful also in the asymptotic analysis of the mixing time of Markov chains, and will be employed in continued work on analysis of the two-dimensional abelian sandpile model and a new local limit theorem on nilpotent groups.