ABSTRACT
We investigate the sequence of random polynomials generated by the three-term recurrence relation
,
, with initial conditions
,
, assuming that
is a sequence of positive i.i.d. random variables.
is a sequence of orthogonal polynomials on the real line, and
is the characteristic polynomial of a Jacobi matrix
. We investigate the relation between the common distribution of the recurrence coefficients
and two other distributions obtained as weak limits of the averaged empirical and spectral measures of
. 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.
MSC 2010:
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.