Scale invariance of the PNG droplet and the Airy process. (English) Zbl 1025.82010

Prähofer, Michael; Spohn, Herbert

J. Stat. Phys. 108, No. 5-6, 1071-1106 (2002).

Summary: 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 J. Baik, P. Deift and K. Johansson [J. Am. Math. Soc. 12, 1119-1178 (1999; Zbl 0932.05001)] on the length of the longest increasing subsequence of a random permutation.

MSC:

82C05	Classical dynamic and nonequilibrium statistical mechanics (general)
82C41	Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics

Keywords:

Airy process; polynuclear growth model; longest increasing subsequences; free fermion techniques