The Transition between the Gap Probabilities from the Pearcey to the Airy Process—a Riemann–Hilbert Approach

We consider the gap probability for the Pearcey and Airy processes; we set up a Riemann–Hilbert approach (different from the standard one) whereby the asymptotic analysis for large gap/large time of the Pearcey process is shown to factorize into two independent Airy processes using the Deift–Zhou steepest descent analysis. Additionally, we relate the theory of Fredholm determinants of integrable kernels and the theory of isomonodromic tau function. Using the Riemann–Hilbert problem mentioned above, we construct a suitable Lax pair formalism for the Pearcey gap probability and re-derive the two nonlinear PDEs recently found and additionally find a third one not reducible to those.