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

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• Average Characteristic Polynomials of Determinantal Point Processes
• Critical measures for vector energy: global structure of trajectories of quadratic differentials
• O'Connell's process as a vicious Brownian motion
• Noncolliding Brownian Motion with Drift and Time-Dependent Stieltjes-Wigert Determinantal Point Process
• Non-intersecting squared Bessel paths at a hard-edge tacnode
• Gap probabilities for the Generalized Bessel process: a Riemann-Hilbert approach
• Multiple orthogonal polynomials, $d$-orthogonal polynomials, production matrices, and branched continued fractions
• Large Deviations for a Non-Centered Wishart Matrix
• Determinantal process starting from an orthogonal symmetry is a Pfaffian process
• Gap probability for the hard edge Pearcey process
• Electrostatic partners and zeros of orthogonal and multiple orthogonal polynomials
• Asymptotics of a discrete-time particle system near a reflecting boundary
• Large n limit for the product of two coupled random matrices
• Critical behavior in Angelesco ensembles
• Multiple Orthogonal Polynomials of two real variables
• Szegő Recurrence for Multiple Orthogonal Polynomials on the Unit Circle