The transition between the gap probabilities from the Pearcey to the Airy process; a Riemann-Hilbert approach

Details The transition between the gap probabilities from the Pearcey to the Airy process; a Riemann-Hilbert approach The transition between the gap probabilities from the Pearcey to the Airy process; a Riemann-Hilbert approach

2010-05-21

arxiv.org/abs/1005.4083v6

Physics

Mathematical Physics

43 pages, 7 figures. Final version with minor changes. Accepted for publication on International Mathematical Research Notices

Scientific paper

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.

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