On random polynomials generated by a symmetric three-term recurrence relation

TL;DR

This paper studies the spectral properties of random orthogonal polynomials generated by a symmetric three-term recurrence with i.i.d. coefficients, revealing combinatorial relations between their spectral measures and moments.

Contribution

It introduces a combinatorial approach to relate the moments of spectral measures of random Jacobi matrices to colored planar trees, expanding understanding of random orthogonal polynomials.

Findings

Distribution of recurrence coefficients influences spectral measures

Moments are characterized by colored planar trees

Spectral measures converge to weak limits

Abstract

We investigate the sequence $(P_{n}(z))_{n=0}^{\infty}$ of random polynomials generated by the three-term recurrence relation $P_{n+1}(z)=z P_{n}(z)-a_{n} P_{n-1}(z)$, $n\geq 1$, with initial conditions $P_{\ell}(z)=z^{\ell}$, $\ell=0, 1$, assuming that $(a_{n})_{n\in\mathbb{Z}}$ is a sequence of positive i.i.d. random variables. $(P_{n}(z))_{n=0}^{\infty}$ is a sequence of orthogonal polynomials on the real line, and $P_{n}$ is the characteristic polynomial of a Jacobi matrix $J_{n}$. We investigate the relation between the common distribution of the recurrence coefficients $a_{n}$ and two other distributions obtained as weak limits of the averaged empirical and spectral measures of $J_{n}$. Our main result is a description of combinatorial relations between the moments of the aforementioned distributions in terms of certain classes of colored planar trees. Our approach is combinatorial, and the starting point of the analysis is a formula of P. Flajolet for weight polynomials associated with labelled Dyck paths.