Non-colliding Brownian bridges and the asymmetric tacnode process

TL;DR

This paper introduces the asymmetric tacnode process, a new determinantal point process describing the limit of non-colliding Brownian bridges touching at a tacnode, generalizing the symmetric case with two parameters.

Contribution

It derives the limiting asymmetric tacnode process with a correlation kernel depending on two parameters, extending previous symmetric tacnode results.

Findings

Defines the asymmetric tacnode process as a determinantal point process.

Provides the correlation kernel parametrized by curvature ratio and interaction parameter.

Generalizes the symmetric tacnode process to an asymmetric setting.

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).