Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights

TL;DR

This paper analyzes a model of non-intersecting squared Bessel paths, revealing their limiting behavior, phase transition at a critical time, and connection to multiple orthogonal polynomials with modified Bessel weights, using Riemann-Hilbert techniques.

Contribution

It introduces a detailed analysis of non-intersecting squared Bessel paths, including phase transition phenomena and their relation to multiple orthogonal polynomials with modified Bessel weights.

Findings

Paths fill a specific region in the plane in the large n limit.

At a critical time, paths hit the hard edge at zero, indicating a phase transition.

Standard kernels like sine, Airy, and Bessel appear in scaling limits.

Abstract

We study a model of $n$ non-intersecting squared Bessel processes in the confluent case: all paths start at time $t = 0$ at the same positive value $x = a$, remain positive, and are conditioned to end at time $t = T$ at $x = 0$. In the limit $n \to \infty$, after appropriate rescaling, the paths fill out a region in the $tx$-plane that we describe explicitly. In particular, the paths initially stay away from the hard edge at $x = 0$, but at a certain critical time $t^*$ the smallest paths hit the hard edge and from then on are stuck to it. For $t \neq t^*$ we obtain the usual scaling limits from random matrix theory, namely the sine, Airy, and Bessel kernels. A key fact is that the positions of the paths at any time $t$ constitute a multiple orthogonal polynomial ensemble, corresponding to a system of two modified Bessel-type weights. As a consequence, there is a $3 \times 3$ matrix valued Riemann-Hilbert problem characterizing this model, that we analyze in the large $n$ limit using the Deift-Zhou steepest descent method. There are some novel ingredients in the Riemann-Hilbert analysis that are of independent interest.