Infinite Random Matrix Theory, Tridiagonal Bordered Toeplitz Matrices, and the Moment Problem

TL;DR

This paper explores the representation of major random matrix theory laws through bordered Toeplitz matrices, compares these laws, and introduces an algorithm for the finite moment problem based on this structure.

Contribution

It introduces a unified bordered Toeplitz matrix framework for major asymptotic laws and proposes a new algorithm for the finite moment problem.

Findings

Unified representation of asymptotic laws using bordered Toeplitz matrices

Comparison and contrast of different random matrix laws

An algorithm for the finite moment problem with bordered Toeplitz density

Abstract

The four major asymptotic level density laws of random matrix theory may all be showcased though their Jacobi parameter representation as having a bordered Toeplitz form. We compare and contrast these laws, completing and exploring their representations in one place. Inspired by the bordered Toeplitz form, we propose an algorithm for the finite moment problem by proposing a solution whose density has a bordered Toeplitz form.