Non-colliding Brownian bridges and the asymmetric tacnode process

Abstract

We consider non-colliding Brownian bridges starting from two points and returning to the same position. These positions are chosen such that, in the limit of large number of bridges, the two families of bridges just touch each other forming a tacnode. We obtain the limiting process at the tacnode, the "asymmetric tacnode process". It is a determinantal point process with correlation kernel given by two parameters: (1) the curvature's ratio $\lambda>0$ of the limit shapes of the two families of bridges, (2) a parameter $\sigma$ controlling the interaction on the fluctuation scale. This generalizes the result for the symmetric tacnode process ($\lambda=1$ case).