Integrable Systems and Random Matrix Theory Seminar

Let $a_n$ be a bi-infinite sequence of positive i.i.d. random variables with all moments finite. In this talk I will consider the sequence of polynomials $P_n(z)$ generated by the recurrence relation $P_{n+1}=z P_n+a_n P_{n-1}$, and analyze the mutual relations between three distributions associated with this sequence, which are the following: the common distribution of the recurrence coefficients a_n, the average zero asymptotic distribution of the polynomials, and the average spectral measure of the associated one-sided Jacobi operator. Our approach is combinatorial, and the starting point is a formula due to P. Flajolet for the Stieltjes-Rogers polynomials. Our main result is a description of the relations between the moments of these three distributions in terms of certain classes of colored planar trees.

If time allows I will discuss some features of the analogous problem in the case of a high-order three-term recurrence relation $P_{n+1}=z P_n+a_{n-r+1} P_{n-r}$, where $r\geq 2$ is a fixed integer. This talk is based on a joint work with V.A. Prokhorov. Speaker(s): Abey Lopez Garcia (University of Central Florida)