### Scale Invariance of the PNG Droplet and the Airy Process

**M. Prähofer, H. Spohn**

J. Stat. Phys. 108, 1071-1106 (2002)

**DOI**: 10.1023/A:1019791415147

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*arXiv.org*: math.PR/0105240

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*MathSciNet*: MR1901953

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**Abstract:**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.