Scale invariance of the PNG droplet and the airy process

Michael Prähofer, Herbert Spohn

Research output: Contribution to journalArticlepeer-review

394 Scopus citations


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 languageEnglish
Pages (from-to)1071-1106
Number of pages36
JournalJournal of Statistical Physics
Issue number5-6
StatePublished - 2002


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


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