Spectral and Dynamical Properties of Certain Random Jacobi Matrices with Growing Parameters

TL;DR

This paper investigates the spectral and dynamical behaviors of a class of random Jacobi matrices with growing off-diagonal terms, revealing operators with fast transport and zero-dimensional spectral measures, with applications to Gaussian beta ensembles.

Contribution

It introduces a novel analysis of random Jacobi matrices with power-law growing parameters, connecting their spectral properties to random decaying potential methods and Gaussian beta ensembles.

Findings

Existence of operators with arbitrarily fast transport

Spectral measures can be zero-dimensional

Application to Gaussian beta ensembles

Abstract

In this paper, a family of random Jacobi matrices, with off-diagonal terms that exhibit power-law growth, is studied. Since the growth of the randomness is slower than that of these terms, it is possible to use methods applied in the study of Schr\"odinger operators with random decaying potentials. A particular result of the analysis is the existence of operators with arbitrarily fast transport whose spectral measure is zero dimensional. The results are applied to the infinite Gaussian $\beta$ Ensembles and their spectral properties are analyzed.