Non-Intersecting Squared Bessel Paths at a Hard-Edge Tacnode

The squared Bessel process is a 1-dimensional diffusion process related to the squared norm of a higher dimensional Brownian motion. We study a model of n non-intersecting squared Bessel paths, with all paths starting at the same point a > 0 at time t = 0 and ending at the same point b > 0 at time t = 1. Our interest lies in the critical regime ab = 1/4, for which the paths are tangent to the hard edge at the origin at a critical time . The critical behavior of the paths for n -> a is studied in a scaling limit with time t = t (*) + O(n (-1/3)) and temperature T = 1 + O(n (-2/3)). This leads to a critical correlation kernel that is defined via a new Riemann-Hilbert problem of size 4 x 4. The Riemann-Hilbert problem gives rise to a new Lax pair representation for the Hastings-McLeod solution to the inhomogeneous Painlev, II equation q''(x) = xq(x) + 2q (3)(x) - nu, where nu = alpha + 1/2 with alpha > -1 the parameter of the squared Bessel process. These results extend our recent work with Kuijlaars and Zhang (Comm Pure Appl Math 64:1305-1383, 2011) for the homogeneous case nu = 0.