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

TL;DR

This paper analyzes the limiting behavior of non-intersecting squared Bessel paths with fixed positive start and end points, revealing phase transitions depending on the product of these points and employing advanced Riemann-Hilbert techniques.

Contribution

It formulates a vector equilibrium problem for the model and connects its solution to the paths' limiting distribution, advancing understanding of phase transitions in such stochastic processes.

Findings

Limiting distribution described by a vector equilibrium problem.

Identification of phase transition locations based on the product ab.

Application of steepest descent analysis to a 4x4 Riemann-Hilbert problem.

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. Depending on the value of the product $ab$ the region may come to the hard edge at 0, or not. 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 $4 \times 4$ 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.