The Tacnode Kernel: Equality of Riemann–Hilbert and Airy Resolvent Formulas

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 for the critical correlation kernel that describes the microscopic behavior of the Brownian motions near the tacnode. One approach, due to Kuijlaars, Zhang, and the author, expresses the kernel via a $$4\times 4$$ matrix-valued Riemann–Hilbert problem (RH problem). Another approach, due to Adler, Ferrari, Johansson, van Moerbeke, and Vetö, expresses the kernel via resolvents and Fredholm determinants of the Airy integral operator. In this paper, we prove the equivalence of both approaches. We also obtain a rank-2 property for the derivative of the tacnode kernel. Finally, we find an RH expression for the multitime extended tacnode kernel.