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

TL;DR

This paper derives explicit formulas for gap probabilities of the tacnode, Pearcey, and Airy point processes using Fredholm determinants, enabling numerical evaluation and analysis of their mutual degenerations.

Contribution

It provides a new explicit Fredholm determinant formula for the tacnode process's gap probabilities, linking it to known distributions and processes.

Findings

Numerical evaluation of gap probabilities across regimes

Demonstration of degenerations from tacnode to Pearcey and Airy processes

Explicit kernel construction with Airy functions and exponentials

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.