Airy process

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. 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 Airy process is named after the Airy function. 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.

There are several Airy processes. The Airy1 process was introduced by Tomohiro Sasomoto and the one-point marginal distribution of the Airy1 is a scalar multiply of the Tracy-Widom distribution of the GOE. Another Airy process is the Airystat process.

Airy2 proces
Let $$t_1<t_2<\dots <t_n$$ be in $$\R$$.

The Airy2 process $$A_2(t)$$ has the following finite-dimensional distribution
 * $$P(A_2(t_{1})<\xi_1,\dots,A_2(t_{n})<\xi_n)=\det(1-f^{1/2}K^{\operatorname{ext}}_{\operatorname{Ai}}f^{1/2})_{L^2(\{t_1,\dots,t_n\}\times \R)}$$

where
 * $$f(t_j,\xi):=1_{\{(\xi_j,\infty)\}}(\xi)$$

and $$K^{\operatorname{ext}}_{\operatorname{Ai}}(t_i,x;t_j,y)$$ is the extended Airy kernel
 * $$K^{\operatorname{ext}}_{\operatorname{Ai}}(t_i,x;t_j,y):=\begin{cases}{\displaystyle \int_0^\infty e^{-z(t_i-t_j )}\operatorname{Ai}(x+z)\operatorname{Ai}(y+z)\mathrm{d}z}& \text{if }\;t_i\geq t_j\\

{\displaystyle -\int_{-\infty}^0 e^{-z(t_i-t_j)}\operatorname{Ai}(x+z)\operatorname{Ai}(y+z)\mathrm{d}z}&\text{if }\;t_i< t_j\end{cases}.$$

Explanations

 * If $$t_i=t_j$$ the extended Airy kernel reduces to the Airy kernel and hence
 * $$P(A_2(t)\leq \xi)=F_{2}(\xi),$$
 * where $$F_{2}(\xi)$$ is the Tracy-Widom distribution of the GUE.


 * $$f^{1/2}K^{\operatorname{ext}}_{\operatorname{Ai}}f^{1/2}$$ is a trace class operator on $$L^2(\{t_1,\dots,t_n\}\times \R)$$ with counting measure on $$\{t_1,\dots,t_n\}$$ and Lebesgue measure on $$\R$$, the kernel is $$f^{1/2}K^{\operatorname{ext}}_{\operatorname{Ai}}(t_i,x;t_j,y)f^{1/2}$$.