Non-colliding Brownian Motions and the extended tacnode process

TL;DR

This paper derives a new extended kernel for non-colliding Brownian motions at a tacnode, revealing detailed correlation structures and extending previous models with novel methods.

Contribution

It introduces a new method to compute the extended kernel for non-colliding Brownian motions at a tacnode, differing from prior formulations.

Findings

Derived the extended kernel at the tacnode point using new techniques.

Obtained the correlation kernel for finite non-colliding Brownian motions with arbitrary endpoints.

Compared the new kernel form with previous models, highlighting differences.

Abstract

We consider non-colliding Brownian motions with two starting points and two endpoints. The points are chosen so that the two groups of Brownian motions just touch each other, a situation that is referred to as a tacnode. The extended kernel for the determinantal point process at the tacnode point is computed using new methods and given in a different form from that obtained for a single time in previous work by Delvaux, Kuijlaars and Zhang. The form of the extended kernel is also different from that obtained for the extended tacnode kernel in another model by Adler, Ferrari and van Moerbeke. We also obtain the correlation kernel for a finite number of non-colliding Brownian motions starting at two points and ending at arbitrary points.