Matthew Lerner-Brecher
MIT
Title
Edge Limits of Random Matrix Ensembles at Zero Temperature
Abstract
The Laguerre beta ensemble, one of the three classical random matrix ensembles, models the behavior of eigenvalues of random covariance matrices. The beta parameter here not only distinguishes between the real, complex, and quaternionic cases (beta = 1,2,4 respectively), but also functions as the inverse temperature when the ensemble is viewed through the lens of interacting particle systems.
In this talk, we will discuss a multilevel extension of the Laguerre beta ensemble at zero temperature (i.e. infinite beta) with emphasis on its hard-edge behavior. In particular, we will see that, as the number of eigenvalues goes to infinity, this behavior can be captured by a discrete time Gaussian process with deep connections to Bessel functions. Our work here builds off Gorin-Kleptsyn (2020), which studied the soft edge behavior of the corresponding Gaussian case, and we will spend some time expounding on interesting connections that arose between our results and theirs.
