## 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.

Original language | English |
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Pages (from-to) | 1071-1106 |

Number of pages | 36 |

Journal | Journal of Statistical Physics |

Volume | 108 |

Issue number | 5-6 |

DOIs | |

State | Published - 2002 |

## Keywords

- Airy process
- Free fermion techniques
- Longest increasing subsequences
- PNG model