The tacnode Riemann-Hilbert problem

TL;DR

This paper studies the tacnode Riemann-Hilbert problem, providing explicit integral representations for all entries of the associated matrix, which helps describe critical phenomena in Brownian motions and two-matrix models.

Contribution

It offers comprehensive integral representations for all entries of the 4x4 matrix in the tacnode RH problem, extending previous work and enabling explicit formulas for the critical kernel.

Findings

Explicit integral representations for all matrix entries.

Derived an explicit formula for the Duits-Geudens critical kernel.

Enhanced understanding of local behavior in Brownian motion and matrix models.

Abstract

The tacnode Riemann-Hilbert problem is a 4 x 4 matrix valued RH problem that appears in the description of the local behavior of two touching groups of non-intersecting Brownian motions. The same RH problem was also found by Duits and Geudens to describe a new critical regime in the two-matrix model. Delvaux gave integral representations for some of the entries of the 4 x 4 matrix. We complement this work by presenting integral representations for all of the entries. As a consequence we give an explicit formula for the Duits-Geudens critical kernel.