Non-intersecting squared Bessel paths with one positive starting and ending point

Abstract

We consider a model of n non-intersecting squared Bessel processes with one starting point a > 0 at time t = 0 and one ending point b > 0 at time t = T. After proper scaling, the paths fill out a region in the tx-plane. The region may come to the hard edge at 0 or may not, depending on the value of the product ab. We formulate a vector equilibrium problem for this model, which is defined for three measures, with upper constraints on the first and third measures and an external field on the second measure. It is shown that the limiting mean distribution of the paths at time t is given by the second component of the vector that minimizes this vector equilibrium problem. The proof is based on a steepest descent analysis for a 4 × 4 matrix-valued Riemann-Hilbert problem which characterizes the correlation kernel of the paths at time t. We also discuss the precise locations of the phase transitions.