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

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 $t^{\ast }$ the smallest paths hit the hard edge and from then on are stuck to it. For $t\ne t^{\ast }$ 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.

59 pages, 11 figures

References in corpus (8)

• The Pearcey Process
• Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems
• Orthogonal polynomial ensembles in probability theory
• Universality of a double scaling limit near singular edge points in random matrix models
• Nonintersecting Brownian excursions
• Noncolliding Brownian Motion and Determinantal Processes
• Two Bessel Bridges Conditioned Never to Collide, Double Dirichlet Series, and Jacobi Theta Function
• Asymptotic analysis of random matrices with external source and a family of algebraic curves