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

* Dedicated to Guillermo López Lagomasino, in celebration of his 70th birthday.

ABSTRACT

We investigate the sequence $\left(P_n\right(z){)}_{n=0}^{\mathrm{\infty }}$ of random polynomials generated by the three-term recurrence relation $P_{n+1}\left(z\right)=z{P}_n\left(z\right)-a_n{P}_{n-1}\left(z\right)$, $n\ge 1$, with initial conditions $P_{\ell }\left(z\right)=z^{\ell }$, $\ell =0,1$, assuming that $(a_n{)}_{n\in \mathbb{Z}}$ is a sequence of positive i.i.d. random variables. $\left(P_n\right(z){)}_{n=0}^{\mathrm{\infty }}$ 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 coloured 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.

KEYWORDS:

• Random polynomial
• orthogonal polynomial
• three-term recurrence relation
• spectral measure
• Dyck path
• coloured tree

MSC 2010:

• Primary 60G55
• 42C05
• Secondary 05C05

Acknowledgements

We thank Jeff Geronimo and Mourad Ismail for bringing to our attention some pertinent works on random polynomials and Jacobi matrices.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

* Dedicated to Guillermo López Lagomasino, in celebration of his 70th birthday.

Additional information

Funding

The first author acknowledges partial support from the grant MTM2015-65888-C4-2-P of the Spanish Ministry of Economy and Competitiveness (Ministerio de Economía y Competitividad, Spain).