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

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 $x = a$,remainpositive,andareconditionedtoendattime$t = T$at$x = 0$. In the limit $n\to \mathrm{\infty }$, 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 $t\mathrm{\ne }{t}^{\ast }$ 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.