The Airy processes are a family of stationary stochastic processes that appear as limit processes in the theory of random growth models and random matrix theory.
They are conjectured to be universal limits describing the long time, large scale spatial fluctuations of the models in the (1+1)-dimensional KPZ universality class (Kardar–Parisi–Zhang equation) for many initial conditions (see also KPZ fixed point).
The original process Airy2 was introduced in 2002 by the mathematicians Michael Prähofer and Herbert Spohn.
[1] They proved that the height function of a model from the (1+1)-dimensional KPZ universality class - the PNG droplet - converges under suitable scaling and initial condition to the Airy2 process and that it is a stationary process with almost surely continuous sample paths.
The process can be defined through its finite-dimensional distribution with a Fredholm determinant and the so-called extended Airy kernel.
It turns out that the one-point marginal distribution of the Airy2 process is the Tracy-Widom distribution of the GUE.
The Airy1 process was introduced by Tomohiro Sasomoto[2] and the one-point marginal distribution of the Airy1 is a scalar multiply of the Tracy-Widom distribution of the GOE.
is the extended Airy kernel