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.^[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 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^[2] and the one-point marginal distribution of the Airy₁ is a scalar multiply of the Tracy-Widom distribution of the GOE.^[3] Another Airy process is the Airy_stat process.^[4]

Airy2 proces

Let ${\displaystyle t_{1}<t_{2}<\dots <t_{n}}$ be in ${\displaystyle \mathbb {R} }$.

The Airy₂ process ${\displaystyle A_{2}(t)}$ has the following finite-dimensional distribution

${\displaystyle P(A_{2}(t_{1})<\xi _{1},\dots ,A_{2}(t_{n})<\xi _{n})=\det(1-f^{1/2}K_{\operatorname {Ai} }^{\operatorname {ext} }f^{1/2})_{L^{2}(\{t_{1},\dots ,t_{n}\}\times \mathbb {R} )}}$

where

${\displaystyle f:=f(t_{j},\xi )=1_{\{(\xi _{j},\infty )\}}(\xi )}$

and ${\displaystyle K_{\operatorname {Ai} }^{\operatorname {ext} }:=K_{\operatorname {Ai} }^{\operatorname {ext} }(t_{i},x;t_{j},y)}$ is the extended Airy kernel

${\displaystyle K_{\operatorname {Ai} }^{\operatorname {ext} }(t_{i},x;t_{j},y):={\begin{cases}{\displaystyle \int _{0}^{\infty }e^{-z(t_{i}-t_{j})}\operatorname {Ai} (x+z)\operatorname {Ai} (y+z)\mathrm {d} z}&{\text{if }}\;t_{i}\geq t_{j},\\{\displaystyle -\int _{-\infty }^{0}e^{-z(t_{i}-t_{j})}\operatorname {Ai} (x+z)\operatorname {Ai} (y+z)\mathrm {d} z}&{\text{if }}\;t_{i}<t_{j}.\end{cases}}}$

Explanations

• If ${\displaystyle t_{i}=t_{j}}$ the extended Airy kernel reduces to the Airy kernel and hence

${\displaystyle P(A_{2}(t)\leq \xi )=F_{2}(\xi ),}$

where ${\displaystyle F_{2}(\xi )}$ is the Tracy-Widom distribution of the GUE.

• ${\displaystyle f^{1/2}K_{\operatorname {Ai} }^{\operatorname {ext} }f^{1/2}}$ is a trace class operator on ${\displaystyle L^{2}(\{t_{1},\dots ,t_{n}\}\times \mathbb {R} )}$ with counting measure on ${\displaystyle \{t_{1},\dots ,t_{n}\}}$ and Lebesgue measure on ${\displaystyle \mathbb {R} }$, the kernel is ${\displaystyle f^{1/2}K_{\operatorname {Ai} }^{\operatorname {ext} }(t_{i},x;t_{j},y)f^{1/2}}$.^[5]

Literature

• Prähofer, Michael; Spohn, Herbert (2002). "Scale Invariance of the PNG Droplet and the Airy Process". Journal of Statistical Physics. 108. Springer. arXiv:math/0105240.
• Johansson, Kurt (2003). "Discrete Polynuclear Growth and Determinantal Processes". Commun. Math. Phys. 242. Springer: 290. arXiv:math/0206208. doi:10.1007/s00220-003-0945-y.
• Tracy, Craig; Widom, Harold (2003). "A System of Differential Equations for the Airy Process". Electron. Commun. Probab. 8: 93–98. arXiv:math/0302033. doi:10.1214/ECP.v8-1074.