Stochastic Surface Growth

Growth phenomena constitute an important field in nonequilibrium statistical mechanics. On the one hand, they are ubiquitous in nature, on the other hand, their theoretical understanding poses a challenging problem for the methods of theoretical physics. Typically, growth processes lead to statistically scale invariant structures. For nonlocal growth, selfsimilar clusters are generated in the form of fractals, as for example snowflakes or patterns on a frosted window. In contrast, surface growth with strictly local rules leads to the formation of a compact body separated by a well-defined surface from its surrounding. In this case the complexity lies in the roughness of the surface generated by the fluctuations of the random attachment. For this type of surface growth, in 1986 Kardar, Parisi, and Zhang (KPZ) proposed a continuum theory, which is defined by the KPZ equation, a nonlinear stochastic partial differential equation. It is arguably the simplest possible equation of motion for the dynamics of an interface, which comprises all the ingredients for nontrivial growth: irreversibility, nonlinearity, stochasticity, and locality. Because of its fundamental importance for nonequilibrium physics the KPZ theory has been and still is the subject of intensive research by means of simulations, field theoretic and other, predominantly approximative methods. In this work an especially simple semi-discrete model, the polynuclear growth (PNG) model, is considered, which lies in the KPZ universality class. Therefore all results for this model obtained in the scaling limit provide direct predictions for the corresponding quantities in KPZ theory and thereby for all models belonging to the same universality class. For growth on a one-dimensional substrate the PNG model is exactly solvable. Through reformulation as a last-passage percolation problem the scaling exponents, predicted by (1 + 1)-dimensional KPZ theory are rigorously derived and for the first time limiting distributions of the surface fluctuations are determined for different growth geometries. Moreover the dynamical KPZ two-point function is expressed by means of the solution to the Riemann-Hilbert problem for the Painlevé II equation and solved numerically, which requires some effort. By means of the extension to a multi-layer model the probability distribution at a given point in time is described by a theory of free fermions on a one-dimensional lattice in Euclidean time. In this formulation the continuum limit is feasible. The fluctuations for curved growth are described by the Airy process [102], introduced for this purpose. Roughly speaking the Airy process corresponds to the trajectory of the last particle in Dyson’s Brownian motion. Closely related to the multi-layer PNG model is the Gates-Westcott model of a vicinal growing surface. The predictions of the corresponding anisotropic KPZ theory are confirmed by an exact solution of the model. Finally Monte-Carlo simulations of the PNG model in higher dimensions are presented.