This research project considers the development of various techniques in the study of small value theory: Both typical behaviors and rare events of the type that positive random quantities take smaller values. The typical behavior deals with the expectation of the minimum over a family of non-negative random variables as the size of the index set grows larger. Topics in this direction include minimum length spanning tree indexed by spanning trees, comparison inequalities for minimum of the absolute value of Gaussian vectors, random assignment type problems indexed by permutations, and the first passage percolation indexed by paths. The rare events of small value type deal with decay probabilities of positive random quantities taking smaller values than typical ones. Topics in this direction include small ball probabilities, lower tail behaviors and level crossing probabilities. The very successful applications of tools developed by the proposer over the last few years will be expanded to a detailed study of various isoperimetric type Gaussian inequalities, tensored random fields, Gaussian chaos, Brownian eddy models, permanental processes, and positivity exponents for random polynomials. The major objective is to build and extend a general small value theory based on systematic study of various methods and diverse applications.
Two fundamental phenomena in probability theory are typical behaviors such as expected values, laws of large numbers and central limit theorems, and rare events such as extremely big or small values. This proposal aims to deepen our understanding of small value phenomena for positive random quantities by developing new techniques in the study of relevant physical and biological random models. This research benefits both undergraduate and graduate education and research. Many open problems and results from the proposed study will be used as students course projects. This research should improve our understanding of extremal random events and provide basic tools for the study of our random environment.