Tridiagonal random matrices, an analytic approach

TL;DR

This paper investigates the eigenvalue distribution of random tridiagonal matrices using an analytical method, relaxing previous assumptions and establishing convergence under second-moment conditions, with additional algebraic insights.

Contribution

It introduces an analytical approach to study eigenvalue distributions of tridiagonal matrices, relaxing assumptions and proving convergence under second-moment conditions.

Findings

Proved convergence of spectral distribution under second-moment assumptions.

Provided an analytical framework for eigenvalue distribution analysis.

Discussed algebraic approaches for complex tridiagonal models.

Abstract

In this paper, we study the limiting distribution of the eigenvalues for random tridiagonal matrix models. The limiting distribution is well described by its moments. Here, an analytical approach allows us, as in the case of Wigner matrices, to relax the assumptions on the random variables. With this method, we proved the convergence of the spectral distribution under an assumption on the second moment. We discuss also about an algebraic approach for the tridiagonal models, which are more complicated than the classic freeness.