Probabilistic Methods in Discrete Structures and Applications

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.