Probabilistic Methods in Discrete Structures and Applications

这是一个发展概率论、解析数论以及这两个领域之间联系的定量结果的项目。概率推理在数论最近的几个突破中发挥了重要作用，包括保罗·埃德·S关于覆盖系统的最小模问题的解决，该问题自20世纪50年代以来一直是公开的。本课题的一个方面是细化分析，研究相关的奇数复盖问题。解析数论中使用的技术可以用来研究统计物理中受马尔科夫链支配的模型。通常，这些技术为计算机模型中观察到的现象提供了严格的证据，有时严格的分析会发现模型预测中的错误，这些错误是由于模型大小有限造成的。该项目旨在将最近关于三次场形状的谱展开的强估计推广到四次和五次场形状的类似谱估计。最小模问题的解决结合了矩方法和概率组合学的技巧，包括Lovasz局部引理和伪随机测量的使用。在开发有关覆盖系统的定量结果时，该项目将应用目前在加法组合学中使用的结构分解技术。振荡积分、驻相方法和范德柯普特方法在解析数论中经常被用来获得精确的估计。这些技巧在马尔可夫链的混合时间的渐近分析中通常也是有用的，并将被用于分析二维阿贝尔沙堆模型和关于幂零群的一个新的局部极限定理。

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.