Stochastic Surface Growth. Dissertation, LMU München

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.      «

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 flu...     »