Airy Process
   HOME

TheInfoList



OR:

The Airy processes are a family of stationary
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
es that appear as limit processes in the theory of
random growth model In common usage, randomness is the apparent or actual lack of definite pattern or predictability in information. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Ind ...
s 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 Herbert Spohn (born 1 November 1946) is a German mathematician and mathematical physicist working in kinetic equations, dynamics of stochastic particle systems, hydrodynamic limit, kinetics of growth processes, disordered systems, open quantum sy ...
. They proved that the height function of a model from the (1+1)-dimensional KPZ universality class - the
PNG droplet Portable Network Graphics (PNG, officially pronounced , colloquially pronounced ) is a raster-graphics file format that supports lossless data compression. PNG was developed as an improved, non-patented replacement for Graphics Interchange ...
- converges under suitable scaling and initial condition to the Airy2 process and that it is a stationary process with
almost surely In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (with respect to the probability measure). In other words, the set of outcomes on which the event does not occur ha ...
continuous sample paths. The Airy process is named after the
Airy function In the physical sciences, the Airy function (or Airy function of the first kind) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function Ai(''x'') and the related function Bi(''x''), are Linear in ...
. The process can be defined through its
finite-dimensional distribution In mathematics, finite-dimensional distributions are a tool in the study of measures and stochastic processes. A lot of information can be gained by studying the "projection" of a measure (or process) onto a finite-dimensional vector space (or fi ...
with a
Fredholm determinant In mathematics, the Fredholm determinant is a complex-valued function which generalizes the determinant of a finite dimensional linear operator. It is defined for bounded operators on a Hilbert space which differ from the identity operator by a ...
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 Tomohiro is a masculine Japanese given name. Written forms Tomohiro can be written using many different combinations of kanji characters. Some examples: *友弘, "friend, vast" *友広, "friend, wide" *友寛, "friend, generosity" *友博, "f ...
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 be in \R. The Airy2 process A_2(t) has the following finite-dimensional distribution :P(A_2(t_)<\xi_1,\dots,A_2(t_)<\xi_n)=\det(1-f^K^_f^)_ where :f:=f(t_j,\xi)=1_(\xi) and K^_:=K^_(t_i,x;t_j,y) is the ''extended Airy kernel'' :K^_(t_i,x;t_j,y):=\begin& \text\;t_i\geq t_j,\\ &\text\;t_i< t_j.\end


Explanations

* If t_i=t_j the extended Airy kernel reduces to the Airy kernel and hence ::P(A_2(t)\leq \xi)=F_(\xi), : where F_(\xi) is the Tracy-Widom distribution of the GUE. * f^K^_f^ is a
trace class operator In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of tra ...
on L^2(\\times \R) with counting measure on \ and Lebesgue measure on \R, the kernel is f^K^_(t_i,x;t_j,y)f^.


Literature

* * *


References

{{Random matrix theory Stochastic processes Statistical mechanics