Polynuclear growth on a flat substrate and edge scaling of GOE eigenvalues

We consider the polynuclear growth (PNG) model in 1+1 dimension with flat initial condition and no extra constraints. Through the Robinson-Schensted-Knuth (RSK) construction, one obtains the multilayer PNG model, which consists of a stack of non-intersecting lines, the top one being the PNG height. The statistics of the lines is translation invariant and at a fixed position the lines define a point process. We prove that for large times the edge of this point process, suitably scaled, has a limit. This limit is a Pfaffian point process and identical to the one obtained from the edge scaling of Gaussian orthogonal ensemble (GOE) of random matrices. Our results give further insight to the universality structure within the KPZ class of 1+1 dimensional growth models.Comment: 40 pages, 6 figures, LaTeX; Section 4 is substantially modifie

Scaling Limit for the Space-Time Covariance of the Stationary Totally Asymmetric Simple Exclusion Process

The totally asymmetric simple exclusion process (TASEP) on the one-dimensional lattice with the Bernoulli \rho measure as initial conditions, 0<\rho<1, is stationary in space and time. Let N_t(j) be the number of particles which have crossed the bond from j to j+1 during the time span [0,t]. For j=(1-2\rho)t+2w(\rho(1-\rho))^{1/3} t^{2/3} we prove that the fluctuations of N_t(j) for large t are of order t^{1/3} and we determine the limiting distribution function F_w(s), which is a generalization of the GUE Tracy-Widom distribution. The family F_w(s) of distribution functions have been obtained before by Baik and Rains in the context of the PNG model with boundary sources, which requires the asymptotics of a Riemann-Hilbert problem. In our work we arrive at F_w(s) through the asymptotics of a Fredholm determinant. F_w(s) is simply related to the scaling function for the space-time covariance of the stationary TASEP, equivalently to the asymptotic transition probability of a single second class particle.Comment: 53 pages, 4 figures, Latex2e; Fixed a numerical prefactor in the scaling function (1.10

Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals

Three models from statistical physics can be analyzed by employing space-time determinantal processes: (1) crystal facets, in particular the statistical properties of the facet edge, and equivalently tilings of the plane, (2) one-dimensional growth processes in the Kardar-Parisi-Zhang universality class and directed last passage percolation, (3) random matrices, multi-matrix models, and Dyson's Brownian motion. We explain the method and survey results of physical interest.Comment: Lecture Notes: Fundamental Problems in Statistical Mechanics XI, Leuven, September 4 - 16, 200

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

Dynamical Correlations among Vicious Random Walkers

Nonintersecting motion of Brownian particles in one dimension is studied. The system is constructed as the diffusion scaling limit of Fisher's vicious random walk. N particles start from the origin at time t=0 and then undergo mutually avoiding Brownian motion until a finite time t=T. In the short time limit $t \ll T$, the particle distribution is asymptotically described by Gaussian Unitary Ensemble (GUE) of random matrices. At the end time t = T, it is identical to that of Gaussian Orthogonal Ensemble (GOE). The Brownian motion is generally described by the dynamical correlations among particles at many times $t_1,t_2,..., t_M$ between t=0 and t=T. We show that the most general dynamical correlations among arbitrary number of particles at arbitrary number of times are written in the forms of quaternion determinants. Asymptotic forms of the correlations in the limit $N \to \infty$ are evaluated and a discontinuous transition of the universality class from GUE to GOE is observed.Comment: REVTeX3.1, 4 pages, no figur

KPZ equation in one dimension and line ensembles

For suitably discretized versions of the Kardar-Parisi-Zhang equation in one space dimension exact scaling functions are available, amongst them the stationary two-point function. We explain one central piece from the technology through which such results are obtained, namely the method of line ensembles with purely entropic repulsion.Comment: Proceedings STATPHYS22, Bangalore, 200

Large time asymptotics of growth models on space-like paths II: PNG and parallel TASEP

We consider the polynuclear growth (PNG) model in 1+1 dimension with flat initial condition and no extra constraints. The joint distributions of surface height at finitely many points at a fixed time moment are given as marginals of a signed determinantal point process. The long time scaling limit of the surface height is shown to coincide with the Airy_1 process. This result holds more generally for the observation points located along any space-like path in the space-time plane. We also obtain the corresponding results for the discrete time TASEP (totally asymmetric simple exclusion process) with parallel update.Comment: 39 pages,6 figure

Exact solution for the stationary Kardar-Parisi-Zhang equation

We obtain the first exact solution for the stationary one-dimensional Kardar-Parisi-Zhang equation. A formula for the distribution of the height is given in terms of a Fredholm determinant, which is valid for any finite time $t$. The expression is explicit and compact enough so that it can be evaluated numerically. Furthermore, by extending the same scheme, we find an exact formula for the stationary two-point correlation function.Comment: 9 pages, 3 figure

Directed diffusion of reconstituting dimers

We discuss dynamical aspects of an asymmetric version of assisted diffusion of hard core particles on a ring studied by G. I. Menon {\it et al.} in J. Stat Phys. {\bf 86}, 1237 (1997). The asymmetry brings in phenomena like kinematic waves and effects of the Kardar-Parisi-Zhang nonlinearity, which combine with the feature of strongly broken ergodicity, a characteristic of the model. A central role is played by a single nonlocal invariant, the irreducible string, whose interplay with the driven motion of reconstituting dimers, arising from the assisted hopping, determines the asymptotic dynamics and scaling regimes. These are investigated both analytically and numerically through sector-dependent mappings to the asymmetric simple exclusion process.Comment: 10 pages, 6 figures. Slight corrections, one added reference. To appear in J. Phys. Cond. Matt. (2007). Special issue on chemical kinetic