  The tacnode process is a universal behavior arising in nonintersecting
particle systems and tiling problems. For Dyson Brownian bridges, the tacnode
process describes the grazing collision of two packets of walkers. We consider
such a Dyson sea on the unit circle with drift. For any integer k, we show that
an appropriate double scaling of the drift and return time leads to a
generalization of the tacnode process in which k particles are expected to wrap
around the circle. We derive winding number probabilities and an expression for
the correlation kernel in terms of functions related to the generalized
Hastings-McLeod solutions to the inhomogeneous Painleve-II equation. The method
of proof is asymptotic analysis of discrete orthogonal polynomials with a
complex weight.
