Scale Invariance of the PNG Droplet and the Airy Process

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

2001-05-29

arxiv.org/abs/math/0105240v3

J. Stat. Phys. 108 (5-6): 1071-1106 (2002)

Mathematics

Probability

32 pages, 1 eps, revised version, the multi-layer dynamics now has two variants, simpler proof of Thm 2.1

Scientific paper

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.

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