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.Wachstumsphänomene stellen ein wichtiges Teilgebiet der statistischen Mechanik des Nichtgleichgewichts dar. Die 1986 von Kardar, Parisi und Zhang (KPZ) vorgeschlagene Kontinuumstheorie sagt für lokales stochastisches Wachstum Skaleninvarianz mit universellen Exponenten und Grenzverteilungen vorher. Für ein spezielles, exakt lösbares, Wachstumsmodell (polynuclear growth - PNG) auf eindimensionalem Substrat (1+1 dimensional) werden die bekannten Skalenexponenten bestätigt und die Grenzverteilungen der Höhenfluktuationen bei verschiedenen Anfangsbedingungen (Tropfen, flach, stationär) erstmals identifiziert. Überraschenderweise sind diese sogenannten Tracy-Widom-Verteilungen aus der Theorie der Zufallsmatrizen bekannt. Die volle stationäre Zweipunkt-Funktion des PNG-Modells wird berechnet. Im Skalenlimes wird sie durch die Lösung eines speziellen Riemann-Hilbert-Problems ausgedrückt und numerisch bestimmt. Auf Grund der erwarteten Universalität erhält man somit eine Vorhersage für die stationäre Zweipunkt-Funktion der (1+1)-dimensionalen KPZ-Theorie. Für die Tropfengeometrie wird gezeigt, dass die Oberflächenfluktuationen im Sinne gemeinsamer Verteilungen gegen den sogenannten Airy-Prozess konvergieren. Schliesslich wird die Theorie für höhere Substratdimensionen diskutiert und durch Monte-Carlo-Simulationen ergänzt

The Airy1_1 process is not the limit of the largest eigenvalue in GOE matrix diffusion

Using a systematic approach to evaluate Fredholm determinants numerically, we provide convincing evidence that the Airy$_1$-process, arising as a limit law in stochastic surface growth, is not the limit law for the evolution of the largest eigenvalue in GOE matrix diffusio

Statistical Self-Similarity of One-Dimensional Growth Processes

For one-dimensional growth processes we consider the distribution of the height above a given point of the substrate and study its scale invariance in the limit of large times. We argue that for self-similar growth from a single seed the universal distribution is the Tracy-Widom distribution from the theory of random matrices and that for growth from a flat substrate it is some other, only numerically determined distribution. In particular, for the polynuclear growth model in the droplet geometry the height maps onto the longest increasing subsequence of a random permutation, from which the height distribution is identified as the Tracy-Widom distribution.Comment: 11 pages, iopart, epsf, 2 postscript figures, submitted to Physica A, in an Addendum the distribution for the flat case is identified analyticall

Fluctuations of an Atomic Ledge Bordering a Crystalline Facet

When a high symmetry facet joins the rounded part of a crystal, the step line density vanishes as sqrt(r) with r denoting the distance from the facet edge. This means that the ledge bordering the facet has a lot of space to meander as caused by thermal activation. We investigate the statistical properties of the border ledge fluctuations. In the scaling regime they turn out to be non-Gaussian and related to the edge statistics of GUE multi-matrix models.Comment: Version with major revisions -- RevTeX, 4 pages, 2 figure

Universal Distributions for Growth Processes in 1+1 Dimensions and Random Matrices

We develop a scaling theory for KPZ growth in one dimension by a detailed study of the polynuclear growth (PNG) model. In particular, we identify three universal distributions for shape fluctuations and their dependence on the macroscopic shape. These distribution functions are computed using the partition function of Gaussian random matrices in a cosine potential.Comment: 4 pages, 3 figures, 1 table, RevTeX, revised version, accepted for publication in PR

Fluctuation properties of the TASEP with periodic initial configuration

We consider the joint distributions of particle positions for the continuous time totally asymmetric simple exclusion process (TASEP). They are expressed as Fredholm determinants with a kernel defining a signed determinantal point process. We then consider certain periodic initial conditions and determine the kernel in the scaling limit. This result has been announced first in a letter by one of us and here we provide a self-contained derivation. Connections to last passage directed percolation and random matrices are also briefly discussed.Comment: 33 pages, 4 figure, LaTeX; We added several references to the general framework and techniques use

Maximum of Dyson Brownian motion and non-colliding systems with a boundary

We prove an equality-in-law relating the maximum of GUE Dyson's Brownian motion and the non-colliding systems with a wall. This generalizes the well known relation between the maximum of a Brownian motion and a reflected Brownian motion