Scale Invariance of the PNG Droplet and the Airy Process

TL;DR

This paper proves that the height fluctuations of the PNG droplet model converge to the Airy process, a universal limit with connections to random matrix theory, using fermionic techniques and multi-layer analysis.

Contribution

It establishes the convergence of the PNG droplet height fluctuations to the Airy process, revealing scale invariance and universality in growth models.

Findings

The Airy process has Tracy-Widom distribution at fixed points.

Height fluctuations exhibit scale invariance under proper rescaling.

Connection established between PNG droplet and Dyson's Brownian motion.

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.