Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On random polynomials generated by a symmetric three-term recurrence
  relation   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.

8
The categories of this paper are: hep-ex
6
The categories of this paper are: astro-ph
30
The categories of this paper are: q-alg
24
The categories of this paper are: dg-ga
27
The categories of this paper are: comp-gas
22
The categories of this paper are: chao-dyn
18
The categories of this paper are: econ
25
The categories of this paper are: patt-sol
32
The categories of this paper are: atom-ph
15
The categories of this paper are: nucl-ex
20
The categories of this paper are: acc-phys
17
The categories of this paper are: q-fin
0
The categories of this paper are: hep-ph
23
The categories of this paper are: solv-int
9
The categories of this paper are: nlin
13
The categories of this paper are: quant-ph