Yun Li: Limits of the truncated circular beta ensembles

Abstract:

Consider the Haar unitary matrix with the first row and column deleted, Życzkowski and Sommers derived the joint distribution of the eigenvalues, and showed that they form a determinantal point process. Killip and Kozhan constructed a family of random matrix models that can be considered as the truncated version of the circular beta ensembles (with \(\beta = 2\) corresponding to the unitary case), and described the spectrum via a random recursion. In this talk, I will discuss the bulk and edge point process limits of the truncated circular beta ensembles, together with the scaling limits of the normalized characteristic polynomials. The limiting objects are closely connected to the hyperbolic Gaussian analytic function and the stochastic zeta function, in the bulk and edge regimes, respectively.

Based on joint work with Benedek Valkó, and ongoing work with Mingchang Liu and Joseph Najnudel.