Maximum of the characteristic polynomial of random Jacobi matrices

TL;DR

This paper analyzes the second order asymptotics of the maximum of the absolute value of the log-characteristic polynomial of certain random Jacobi matrices, providing partial confirmation of a conjecture related to Gaussian beta ensembles.

Contribution

It computes the second order asymptotics of the maximum of the log-characteristic polynomial for random Jacobi matrices, advancing understanding of their extremal properties.

Findings

Partial confirmation of Fydorov-Simm conjecture

Second order asymptotics of the maximum established

Results apply to matrices with exponential integrability conditions

Abstract

We compute the second order asymptotics of the maximum of the absolute value of the log-characteristic polynomial of random Jacobi matrices whose coefficients satisfy some exponential integrability condition. In particular, by the triadiagonal representation of Dumitriu and Eldelman of Gaussian $\beta$ Ensembles, this result partially confirms the Fydorov-Simm conjecture.