Asymptotic analysis of recurrence coefficients coming from non-intersecting squared Bessel paths

P. ROMÁN; A. KUIJLAARS

Coimbra, Portugal

Workshop; 11th European Workshop on Applications and Generalizations of Complex Analysis; 2009

CMUC-Centro de Matemática da Universidade de Coimbra

In this work we consider a model of $n$ non-intersecting squared Bessel paths conditioned so that all paths start at time $t=0$ at the same positive value $a=0$, remain positive and end at time $t=T$ at $x=0$. The positions of the paths at any time form a multiple orthogonal polynomial ensemble corresponding to a system of two modified Bessel-type weights. The type II multiple orthogonal polynomials associated with these weights satisfy a four term recurrence relation. We associate to it a family of $n\times n$ banded Toeplitz matrices $\{ T^s_n \}_{s\geq 0}$. The limiting normalized eigenvalue distribution of $T_n^s$, $\mu_0^s$, is the first component of $(\mu_0^s,\mu_1^s)$ the unique vector of measures that minimizes an energy functional. By integrating the Euler-Lagrange equations for the equilibrium problem for $(\mu_0,\mu_1)$, we obtain an equilibrium problem for $(\nu_0,\nu_1)$,$$\nu_0=\int_0^1 \mu_0^s ds, \quad \nu_1=\int_0^1 \mu_1^s ds.$$This is an equilibrium problem with an external field, and an upper constraint acting on $\nu_1$.