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.

💡 Research Summary

The paper investigates the transition of gap probabilities from the Pearcey process to the Airy process by formulating the problem in terms of a Riemann–Hilbert (RH) representation that differs from the conventional approach used for integrable kernels. The authors begin by expressing the Pearcey kernel, which defines a determinantal point process, in the integrable form introduced by Its, Izergin, Korepin, and Slavnov. This representation allows the Fredholm determinant associated with the Pearcey gap probability to be recast as the solution of a 2 × 2 matrix RH problem with a jump matrix built from two vector functions. The RH problem is normalized at infinity and has its jump contour chosen to reflect the natural Stokes geometry of the Pearcey kernel.

The core of the analysis is the application of the Deift–Zhou steepest descent method to this RH problem in the double‑scaling regime where both the gap endpoint (s) and the Pearcey time parameter (\tau) tend to infinity in a correlated way. By a careful deformation of the contour and the construction of local parametrices near the critical points, the authors show that the original Pearcey RH problem asymptotically splits into two independent Airy RH problems. Each Airy model is solved explicitly in terms of Airy functions, and the matching with the global solution yields an error term of order (\mathcal{O}(\tau^{-1})). Consequently, the Pearcey gap probability factorizes into the product of two Airy gap probabilities up to negligible corrections: \