Stochastic Surface Growth

Growth phenomena constitute an important field in nonequilibrium
statistical mechanics. Kardar, Parisi, and Zhang (KPZ) in 1986
proposed a continuum theory for local stochastic growth predicting
scale invariance with universal exponents and limiting
distributions. For a special, exactly solvable growth model
(polynuclear growth - PNG) on a one-dimensional substrate (1+1
dimensional) we confirm the known scaling exponents and identify
for the first time the limiting distributions of height
fluctuations for different initial conditions (droplet, flat,
stationary). Surprisingly, these so-called Tracy-Widom
distributions have been encountered earlier in random matrix
theory. The full stationary two-point function of the PNG model is
calculated. Its scaling limit is expressed in terms of the solution
to a special Rieman-Hilbert problem and determined numerically. By
universality this yields a prediction for the stationary two-point
function of (1+1)-dimensional KPZ theory. For the PNG droplet we
show that the surface fluctuations converge to the so-called Airy
process in the sense of joint distributions. Finally we discuss the
theory for higher substrate dimensions and provide some Monte-Carlo
simulations.