The tacnode kernel: equality of Riemann-Hilbert and Airy resolvent formulas

Understand

We study nonintersecting Brownian motions with two prescribed starting and ending positions, in the neighborhood of a tacnode in the time-space plane.

• Several expressions have been obtained in the literature for the critical correlation kernel $K\tac(x,y)$ that describes the microscopic behavior of the Brownian motions near the tacnode.
• One approach, due to Kuijlaars, Zhang and the author, expresses the kernel (in the single time case) in terms of a $4\times 4$ matrix valued Riemann-Hilbert problem.
• Another approach, due to Adler, Ferrari, Johansson, van Moerbeke and Vet\H o in a series of papers, expresses the kernel in terms of resolvents and Fredholm determinants of the Airy integral operator acting on a semi-infinite interval $[\sigma,\infty)$, involving some objects introduced by Tracy and Widom.