Scale Invariance of the PNG Droplet and the Airy Process

By Michael Praehofer, Herbert Spohn. 2001

635 citations recorded by the source corpus.

We establish that the static height fluctuations of a particular growth model, the PNG droplet, converges upon proper rescaling to a limit process, which we call the Airy process A(y). The Airy process is stationary, it has continuous sample paths, its single “time” (fixed y) distribution is the Tracy–Widom distribution of the largest eigenvalue of a GUE random matrix, and the Airy process has a slow decay of correlations as y−2. Roughly the Airy process describes the last line of Dyson's Brownian motion model for random matrices. Our construction uses a multi-layer version of the PNG model, which can be analyzed through fermionic techniques. Specializing our result to a fixed value of y, one reobtains the celebrated result of Baik, Deift, and Johansson on the length of the longest increasing subsequence of a random permutation.

Subfields: Mathematical Physics, Molecular Biology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/40682704b630e2a099770e8d25c0f58db19126f4

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Citations in the corpus

Exact Scaling Functions for One-Dimensional Stationary KPZ Growth (2002, methodology): “The factor 2 in the argument of the numerator differs from the convention used by Baik and Rains [12] by a factor 2 but conforms with the definition of the closely related Airy process [7] and has the further advantage to absorb a lot of prefactors in the equations defining g(y).”
Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions (2010, methodology): “→ A2(X) whereA2(X) is the Airy2 process (see [24]).”
The One-Dimensional KPZ Equation and Its Universality Class (2015, background): “More generally one expects that w 7→ γ t η(2γ(2) tw, t) as a stochastic process converges to the Airy process [38], defined through finite dimensional distributions given by expressions generalizing (2.”
The Kardar-Parisi-Zhang equation and universality class (2011, methodology): “The formula was independently and in parallel discovered in mathematics by Amir-Corwin-Quastel [5] and mathematical physics by Sasamoto-Spohn [147, 148, 149] in the fall of 2009 and posted early in 2010.”
Evidence for Geometry-Dependent Universal Fluctuations of the Kardar-Parisi-Zhang Interfaces in Liquid-Crystal Turbulence (2012, background): “Theoretical studies have shown that in the asymptotic limit the spatial correlation of the interface fluctuations is given exactly by the time correlation of the stochastic process called the Airy2 process A2(t ′) for the curved interfaces [73, 75] and the Airy1 process A1(t ′) for”
Nonintersecting Brownian Excursions (2006, background): “The top path Xn(τ), again suitably rescaled, we expect to have the Airy process [13, 20] as its distributional limit.”
Random Skew Plane Partitions and the Pearcey Process (2005, background): “2. at a general point of the frozen boundary, suitably scaled, the correlation are given by the extended Airy kernel [ 17 , 9, 19]; 3. at a cusp of the frozen boundary, correlation, suitably scaled, are given by the extended”
An appetizer to modern developments on the Kardar–Parisi–Zhang universality class (2017, background): “Analytic expressions for such limiting processes have been obtained for the the circular and flat subclasses [70], and named the Airy2 processA2(ζ) for the circular case [66, 97] and the Airy1 processA1(ζ) for the flat case [83, 98].”
Scaling Limit for the Space-Time Covariance of the Stationary Totally Asymmetric Simple Exclusion Process (2005, methodology): “The technique of line ensembles can be used also for the investigation of multipoint statistics [20, 9], which however will not be needed in our context.”
The directed landscape (2018, background): “This next theorem was proven in many parts, see Prähofer and Spohn (2002), Johansson (2003), Adler and Van Moerbeke (2005), Corwin and Hammond (2014).”
Height Fluctuations for the Stationary KPZ Equation (2014, result): “esses forthese models were determined. These statistical properties agreed between the models, but depended non-trivially on the type of initial data or geometry for the growth models, such as curved [6,10,21,22,60,61,68], ﬂat [10,21,23–25,43,48,73] or stationary [7–9,47,49,57,69]. All these results”
Discrete Polynuclear Growth and Determinantal Processes (2002, methodology): “2M−1 ∫ (Rn)2M−1 det(φ−M,−M+1(x −M i , x −M+1 j ))1≤i,j≤n × M−1 ∏ r=−M+1 det((1 − g(r, xri ))φr,r+1(xri , xr+1j ))1≤i,j≤ndµ(x̄) = det”
The KPZ fixed point (2016, methodology): “path integral formula (see AppendixA.2for the proof), det I K(nm) t (I Qn 1 nm˜ a 1 Q n 2 n 1˜ a 2 Q nm n m 1˜ am ) L (R) ; (2.9) where K(n) t = K t(n;;n;): (2.10) Such formulas were ﬁrst obtained in [PS02] for the Airy 2 process (see [PS11] for the proof), and later extended to the Airy 1 process i”
Non-equilibrium steady states: fluctuations and large deviations of the density and of the current (2007, background): “Concerning the fluctuations or the large deviations of the current of driven diffusive systems, there has been lots of progress over the last ten years [133, 134, 135, 136, 137, 138].”
Stochastic six-vertex model (2014, methodology): “ly, we are working with a special class of the asymmetric six-vertex models STOCHASTIC SIX-VERTEX MODEL 3 whose parameters fall on what is sometimes called the stochastic line, cf. [GS], [ADW], [Ki], [PS2]. More formally, we introduce and study the family of measures P(b 1;b 2) depending on two para”
A pedestrian's view on interacting particle systems, KPZ universality and random matrices (2008, background): “4), who named it the Airy process [102].”