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

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.