A Study on Fredholm Determinants Associated with Finite Temperature Kernels

Description

Determinantal point processes are important stochastic models in both mathematics and physics. They are closely related to numerous research topics, such as random matrix theory, Dyson Brownian motion, free fermionic theory, quantum gravity, etc. A central objective in the study of determinantal point processes is to understand the probability that there are exactly certain number of points in a given interval. It is well-known that this probability can be expressed in terms of Fredholm determinants of integral operators with corresponding correlation kernels. Based on this remarkable relation, various significant results have been obtained in the literature, such as explicit expressions of the probability, the large gap asymptotics, the rigidity bounds, etc. In recent years, motivated by quantum mechanics and other interesting models, there are a considerable amount of research interests in Fredholm determinants associated with kernels under finite temperature deformations. Although breakthrough has been made, there are still many significant problems unsolved. In this project, we plan to study Fredholm determinants associated with finite-temperature kernels using matrix-valued and operator-valued Riemann-Hilbert problems. We will derive the explicit expressions, and establish their relation with integrable systems. Furthermore, we will try to derive large gap asymptotics of these Fredholm determinants under various scaling limits, when width of the gap tends to infinity.