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

Abstract

We consider the gap probability for the Pearcey and Airy processes; we set up\na Riemann--Hilbert approach (different from the standard one) whereby the\nasymptotic analysis for large gap/large time of the Pearcey process is shown to\nfactorize into two independent Airy processes using the Deift-Zhou steepest\ndescent analysis. Additionally we relate the theory of Fredholm determinants of\nintegrable kernels and the theory of isomonodromic tau function. Using the\nRiemann-Hilbert problem mentioned above we construct a suitable Lax pair\nformalism for the Pearcey gap probability and re-derive the two nonlinear PDEs\nrecently found and additionally find a third one not reducible to those.\n