Non-colliding Brownian bridges and the asymmetric tacnode process

Patrik Ferrari,Bálint Vető

doi:10.1214/ejp.v17-1811

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).

Highlights

Introduction and resultsSystems of non-colliding Brownian motions have been much studied recently
They arise in random matrix theory, as limit processes of random walk, discrete growth models, and random tiling problems, see e.g. [9,10,11,12,13, 16, 21, 23, 24]
Assume that the starting and ending points are chosen such that in the limit of large number of bridges occupy a region bordered by a deterministic limit shape

Summary

Introduction and results

Systems of non-colliding Brownian motions have been much studied recently. They arise in random matrix theory (see e.g. [17,18,20]), as limit processes of random walk, discrete growth models, and random tiling problems, see e.g. [9,10,11,12,13, 16, 21, 23, 24]. We consider (1 + λ)n non-colliding standard Brownian motions with two starting points and two endpoints where λ > 0 is a fixed parameter. The (asymmetric) tacnode process T σ,λ obtained by the limit of the two non-colliding families of n respectively λn Brownian motions under the scaling (1)–(4) in the neighborhood of the tacnode is given by the following gap probabilities. The (asymmetric) tacnode process has an intrinsic symmetry under the reflection on the horizontal axis that is inherited from the finite system of Brownian motions This corresponds to the following transformation of the variables:. Using this definition, we can give another expression for the kernel Lλta,σc which is formally similar to (5), but the ingredients can be given by a single integral as follows

Johansson’s formula

Proof of the main results

Asymptotic analysis