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

In this paper we consider the model of nn non-intersecting squared Bessel processes with parameter αα, in the confluent case where all particles start, at time t=0t=0, at the same positive value x=ax=a, remain positive, and end, at time T=tT=t, at the position x=0x=0. The positions of the paths have a limiting mean density as n→∞n→∞ 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 αα, letting it increase proportionally to nn as nn increases. In this case we also analyze the recurrence relation and obtain a vector equilibrium problem for it.