Recurrence relations and vector equilibrium problems arising from a model of non-intersecting squared Bessel paths

TL;DR

This paper studies non-intersecting squared Bessel paths, deriving vector equilibrium problems from recurrence relations of associated orthogonal polynomials, including a case where the parameter scales with the number of paths.

Contribution

It introduces a novel connection between recurrence relations of multiple orthogonal polynomials and vector equilibrium problems for non-intersecting Bessel paths, including a parameter-scaling extension.

Findings

Derived vector equilibrium problem from recurrence relations.

Extended analysis to parameter scaling with path number.

Provided asymptotic density characterization for the paths.

Abstract

In this paper we consider the model of $n$ non-intersecting squared Bessel processes with parameter $\alpha$, in the confluent case where all particles start, at time $t=0$, at the same positive value $x=a$, remain positive, and end, at time $T=t$, at the position $x=0$. The positions of the paths have a limiting mean density as $n\to\infty$ which is characterized by a vector equilibrium problem. We show how to obtain this equilibrium problem from different considerations involving the recurrence relations for multiple orthogonal polynomials associated with the modified Bessel functions. We also extend the situation by rescaling the parameter $\alpha$, letting it increase proportionally to $n$ as $n$ increases. In this case we also analyze the recurrence relation and obtain a vector equilibrium problem for it.