Infinite random matrix theory, tridiagonal bordered Toeplitz matrices, and the moment problem

The four major asymptotic level density laws of random matrix theory may all be showcased through 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.