  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.
