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。概率推理在最近的几次数论突破中发挥了重要作用，包括Paul Erdös覆盖系统的最小模问题的解决，该问题自20世纪50年代以来一直开放。本课题的一个方面是细化分析，研究相关的奇覆盖问题。在分析数论中使用的技术可以用来研究统计物理中由马尔可夫链控制的模型。通常，这些技术对计算机模型中观察到的现象提供严格的证明，有时，严格的分析揭示了由于模型大小有限而导致的模型预测错误。该项目旨在将最近对立方场形状的谱展开的强估计扩展到类似的四次和五次场形状的谱估计。最小模问题的解结合了矩法和概率组合学的技术，包括Lovasz局部引理和伪随机测度的使用。在开发关于覆盖系统的定量结果时，该项目将应用目前在加性组合学中使用的结构分解技术。在解析数论中经常使用振荡积分法、定相法和范德科普特法来获得尖锐估计。这些技术在马尔可夫链混合时间的渐近分析中也很有用，并将在二维阿贝尔沙堆模型的分析和幂零群的一个新的局部极限定理的后续工作中得到应用。

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.