The gap probabilities of the tacnode, Pearcey and Airy point processes, their mutual relationship and evaluation

Abstract

We express the gap probabilities of the tacnode process as the ratio of two Fredholm determinants; the denominator is the standard Tracy-Widom distribution, while the numerator is the Fredholm determinant of a very explicit kernel constructed with Airy functions and exponentials. The formula allows us to apply the theory of numerical evaluation of Fredholm determinants and thus produce numerical results for the gap probabilities. In particular we investigate numerically how, in different regimes, the Pearcey process degenerates to the Airy one, and the tacnode degenerates to the Pearcey and Airy ones.

17 pages, 4 figures, final version accepted for publication on Random Matrices: Theory and Applications (RMTA)

References in corpus (4)

• The Pearcey Process
• Non-colliding Brownian bridges and the asymmetric tacnode process
• Riemann-Hilbert approach to multi-time processes; the Airy and the Pearcey case
• From the Pearcey to the Airy process

Cited by in corpus (4)

• The hard edge tacnode process and the hard edge Pearcey process with non-intersecting squared Bessel paths
• Asymptotics of the Tacnode process: a transition between the gap probabilities from the Tacnode to the Airy process
• Integrable structure of products of finite complex Ginibre random matrices
• On the zeros of the Pearcey integral and a Rayleigh-type equation