Non-intersecting squared Bessel paths: critical time and double scaling limit

We consider the double scaling limit for 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 $t=1$ at $x=0$. After appropriate rescaling, the paths fill a region in the $tx$--plane as $n\to \infty$ that intersects the hard edge at $x=0$ at a critical time $t=t^{*}$. In a previous paper (arXiv:0712.1333), the scaling limits for the positions of the paths at time $t\neq t^{*}$ were shown to be the usual scaling limits from random matrix theory. Here, we describe the limit as $n\to \infty$ of the correlation kernel at critical time $t^{*}$ and in the double scaling regime. We derive an integral representation for the limit kernel which bears some connections with the Pearcey kernel. The analysis is based on the study of a $3\times 3$ matrix valued Riemann-Hilbert problem by the Deift-Zhou steepest descent method. The main ingredient is the construction of a local parametrix at the origin, out of the solutions of a particular third-order linear differential equation, and its matching with a global parametrix.