2001/05/31 by Michael Praehofer, Herbert Spohn · 11 citations
Mathematics · Physics and Astronomy · #math.PR #cond-mat.stat-mech
published as J. Stat. Phys. 108 (5-6): 1071-1106 (2002) · 32 pages, 1 eps, revised version, the multi-layer dynamics now has two variants, simpler proof of Thm 2.1
arxiv created 2002/04/24 · arxiv updated 2009/11/30
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.