Solvable Families of Random Block Tridiagonal Matrices

TL;DR

This paper introduces two new families of random block tridiagonal matrices with explicitly computable eigenvalue distributions, revealing novel interactions and describing their spectral limits via differential operators and diffusions.

Contribution

It presents novel random matrix models with explicit eigenvalue distributions and analyzes their spectral edge limits using differential operators and coupled diffusions.

Findings

Eigenvalue distributions can be computed explicitly for the new matrix families.

Spectral edge limits are described via random differential operators.

Algebraic identities involving Vandermonde sums are established.

Abstract

We introduce two families of random tridiagonal block matrices for which the joint eigenvalue distributions can be computed explicitly. These distributions are novel within random matrix theory, and exhibit interactions among eigenvalue coordinates beyond the typical mean-field log-gas type. Leveraging the matrix models, we go on to describe the point process limits at the edges of the spectrum in two ways: through certain random differential operators, and also in terms of coupled systems of diffusions. Along the way we establish several algebraic identities involving sums of Vandermonde determinant products.