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.{{cite journal|first1=Michael|last1=Prähofer|first2=Herbert|last2=Spohn |title=Scale Invariance of the PNG Droplet and the Airy Process |journal=Journal of Statistical Physics|publisher=Springer |volume=108 |date=2002 |arxiv=math/0105240}} 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{{cite journal|doi=10.1088/0305-4470/38/33/l01|date=2005|publisher=IOP Publishing|volume=38|number=33|pages=L549-L556|first1=Tomohiro|last1=Sasamoto|title=Spatial correlations of the 1D KPZ surface on a flat substrate|journal=Journal of Physics A: Mathematical and General|arxiv=cond-mat/0504417}} and the one-point marginal distribution of the Airy₁ is a scalar multiply of the Tracy-Widom distribution of the GOE.{{cite journal|first1=Riddhipratim|last1=Basu|first2=Patrick L.|last2=Ferarri|title=On the Exponent Governing the Correlation Decay of the Airy1 Process |journal=Commun. Math. Phys.|publisher=Springer |date=2022 |doi=10.1007/s00220-022-04544-1|arxiv=2206.08571}} Another Airy process is the Airy_stat process.{{cite journal|doi=10.1002/cpa.20316|date=2010|publisher = Wiley|first1=Jinho|last1=Baik|first2=Patrik L.|last2=Ferrari|first3=Sandrine|last3=Péché |title=Limit process of stationary TASEP near the characteristic line|journal=Communications on Pure and Applied Mathematics|volume=63|number=8|pages=1017–1070|hdl=2027.42/75781|hdl-access=free|arxiv=0907.0226}}

Airy<sub>2</sub> proces

Let $t_1 be in \R .$

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

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

and $K^{\operatorname{ext}}_{\operatorname{Ai}}:=K^{\operatorname{ext}}_{\operatorname{Ai}}(t_i,x;t_j,y)$ 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 $f^{1/2}K^{\operatorname{ext}}_{\operatorname{Ai}}(t_i,x;t_j,y)f^{1/2}$ .{{cite journal|first1=Kurt|last1=Johansson |title=Discrete Polynuclear Growth and Determinantal Processes |journal=Commun. Math. Phys. |volume=242 |date=2003 |page=290 |arxiv=math/0206208|publisher=Springer |doi=10.1007/s00220-003-0945-y}}

Literature

{{cite journal|first1=Michael|last1=Prähofer|first2=Herbert|last2=Spohn |title=Scale Invariance of the PNG Droplet and the Airy Process |journal=Journal of Statistical Physics|publisher=Springer |volume=108 |date=2002 |arxiv=math/0105240}}
{{cite journal|first1=Kurt|last1=Johansson |title=Discrete Polynuclear Growth and Determinantal Processes |journal=Commun. Math. Phys. |volume=242 |date=2003 |page=290 |arxiv=math/0206208|publisher=Springer |doi=10.1007/s00220-003-0945-y}}
{{cite journal|first1=Craig|last1=Tracy|first2=Harold|last2=Widom|title=A System of Differential Equations for the Airy Process.|journal=Electron. Commun. Probab.|volume=8|pages=93–98|date=2003|doi=10.1214/ECP.v8-1074|arxiv=math/0302033}}

References

Category:Stochastic processes

Category:Statistical mechanics