A CLT for the characteristic polynomial of random Jacobi matrices, and the G$\beta$E

TL;DR

This paper establishes a central limit theorem for the logarithm of the characteristic polynomial of random Jacobi matrices, including the GβE models, providing new probabilistic insights into their spectral properties.

Contribution

It introduces a CLT for the characteristic polynomial of random Jacobi matrices, extending to GβE models for all positive β, a significant advancement in spectral theory.

Findings

Proves a CLT for the log of the characteristic polynomial.

Applies results specifically to GβE models for β>0.

Enhances understanding of spectral fluctuations in random matrices.

Abstract

We prove a central limit theorem for the logarithm of the characteristic polynomial of random Jacobi matrices. Our results cover the G$\beta$E models for $\beta>0$.