Quantized Scaling of Growing Surfaces

The Kardar-Parisi-Zhang universality class of stochastic surface growth is studied by exact field-theoretic methods. From previous numerical results, a few qualitative assumptions are inferred. In particular, height correlations should satisfy an operator product expansion and, unlike the correlations in a turbulent fluid, exhibit no multiscaling. These properties impose a quantization condition on the roughness exponent $\chi$ and the dynamic exponent $z$. Hence the exact values $\chi = 2/5, z = 8/5$ for two-dimensional and $\chi = 2/7, z = 12/7$ for three-dimensional surfaces are derived.Comment: 4 pages, revtex, no figure

Upper critical dimension, dynamic exponent and scaling functions in the mode-coupling theory for the Kardar-Parisi-Zhang equation

We study the mode-coupling approximation for the KPZ equation in the strong coupling regime. By constructing an ansatz consistent with the asymptotic forms of the correlation and response functions we determine the upper critical dimension d_c=4, and the expansion z=2-(d-4)/4+O((4-d)^2) around d_c. We find the exact z=3/2 value in d=1, and estimate the values 1.62, 1.78 for z, in d=2,3. The result d_c=4 and the expansion around d_c are very robust and can be derived just from a mild assumption on the relative scale on which the response and correlation functions vary as z approaches 2.Comment: RevTex, 4 page

Controlling surface statistical properties using bias voltage: Atomic force microscopy and stochastic analysis

The effect of bias voltages on the statistical properties of rough surfaces has been studied using atomic force microscopy technique and its stochastic analysis. We have characterized the complexity of the height fluctuation of a rough surface by the stochastic parameters such as roughness exponent, level crossing, and drift and diffusion coefficients as a function of the applied bias voltage. It is shown that these statistical as well as microstructural parameters can also explain the macroscopic property of a surface. Furthermore, the tip convolution effect on the stochastic parameters has been examined.Comment: 8 pages, 11 figures

Directed polymers and interfaces in random media : free-energy optimization via confinement in a wandering tube

We analyze, via Imry-Ma scaling arguments, the strong disorder phases that exist in low dimensions at all temperatures for directed polymers and interfaces in random media. For the uncorrelated Gaussian disorder, we obtain that the optimal strategy for the polymer in dimension $1+d$ with $0<d<2$ involves at the same time (i) a confinement in a favorable tube of radius $R_S \sim L^{\nu_S}$ with $\nu_S=1/(4-d)<1/2$ (ii) a superdiffusive behavior $R \sim L^{\nu}$ with $\nu=(3-d)/(4-d)>1/2$ for the wandering of the best favorable tube available. The corresponding free-energy then scales as $F \sim L^{\omega}$ with $\omega=2 \nu-1$ and the left tail of the probability distribution involves a stretched exponential of exponent $\eta= (4-d)/2$. These results generalize the well known exact exponents $\nu=2/3$, $\omega=1/3$ and $\eta=3/2$ in $d=1$, where the subleading transverse length $R_S \sim L^{1/3}$ is known as the typical distance between two replicas in the Bethe Ansatz wave function. We then extend our approach to correlated disorder in transverse directions with exponent $\alpha$ and/or to manifolds in dimension $D+d=d_{t}$ with $0<D<2$. The strategy of being both confined and superdiffusive is still optimal for decaying correlations ($\alpha<0$), whereas it is not for growing correlations ($\alpha>0$). In particular, for an interface of dimension $(d_t-1)$ in a space of total dimension $5/3<d_t<3$ with random-bond disorder, our approach yields the confinement exponent $\nu_S = (d_t-1)(3-d_t)/(5d_t-7)$. Finally, we study the exponents in the presence of an algebraic tail $1/V^{1+\mu}$ in the disorder distribution, and obtain various regimes in the $(\mu,d)$ plane.Comment: 19 page

Critical dimensions of the diffusion equation

We study the evolution of a random initial field under pure diffusion in various space dimensions. From numerical calculations we find that the persistence properties of the system show sharp transitions at critical dimensions d1 ~ 26 and d2 ~ 46. We also give refined measurements of the persistence exponents for low dimensions.Comment: 4 pages, 5 figure

Width Distributions and the Upper Critical Dimension of KPZ Interfaces

Simulations of restricted solid-on-solid growth models are used to build the width-distributions of d=2-5 dimensional KPZ interfaces. We find that the universal scaling function associated with the steady-state width-distribution changes smoothly as d is increased, thus strongly suggesting that d=4 is not an upper critical dimension for the KPZ equation. The dimensional trends observed in the scaling functions indicate that the upper critical dimension is at infinity.Comment: 4 pages, 4 postscript figures, RevTe

Growing interfaces uncover universal fluctuations behind scale invariance

Stochastic motion of a point -- known as Brownian motion -- has many successful applications in science, thanks to its scale invariance and consequent universal features such as Gaussian fluctuations. In contrast, the stochastic motion of a line, though it is also scale-invariant and arises in nature as various types of interface growth, is far less understood. The two major missing ingredients are: an experiment that allows a quantitative comparison with theory and an analytic solution of the Kardar-Parisi-Zhang (KPZ) equation, a prototypical equation for describing growing interfaces. Here we solve both problems, showing unprecedented universality beyond the scaling laws. We investigate growing interfaces of liquid-crystal turbulence and find not only universal scaling, but universal distributions of interface positions. They obey the largest-eigenvalue distributions of random matrices and depend on whether the interface is curved or flat, albeit universal in each case. Our exact solution of the KPZ equation provides theoretical explanations.Comment: 5 pages, 3 figures, supplementary information available on Journal pag

Probability distribution of the free energy of a directed polymer in a random medium

We calculate exactly the first cumulants of the free energy of a directed polymer in a random medium for the geometry of a cylinder. By using the fact that the n-th moment of the partition function is given by the ground state energy of a quantum problem of n interacting particles on a ring of length L, we write an integral equation allowing to expand these moments in powers of the strength of the disorder gamma or in powers of n. For n small and n of order (L gamma)^(-1/2), the moments take a scaling form which allows to describe all the fluctuations of order 1/L of the free energy per unit length of the directed polymer. The distribution of these fluctuations is the same as the one found recently in the asymmetric exclusion process, indicating that it is characteristic of all the systems described by the Kardar-Parisi-Zhang equation in 1+1 dimensions.Comment: 18 pages, no figure, tu appear in PRE 61 (2000

Derivation of continuum stochastic equations for discrete growth models

We present a formalism to derive the stochastic differential equations (SDEs) for several solid-on-solid growth models. Our formalism begins with a mapping of the microscopic dynamics of growth models onto the particle systems with reactions and diffusion. We then write the master equations for these corresponding particle systems and find the SDEs for the particle densities. Finally, by connecting the particle densities with the growth heights, we derive the SDEs for the height variables. Applying this formalism to discrete growth models, we find the Edwards-Wilkinson equation for the symmetric body-centered solid-on-solid (BCSOS) model, the Kardar-Parisi-Zhang equation for the asymmetric BCSOS model and the generalized restricted solid-on-solid (RSOS) model, and the Villain--Lai--Das Sarma equation for the conserved RSOS model. In addition to the consistent forms of equations for growth models, we also obtain the coefficients associated with the SDEs.Comment: 5 pages, no figur

Effect of Long-Range Interactions in the Conserved Kardar-Parisi-Zhang Equation

The conserved Kardar-Parisi-Zhang equation in the presence of long-range nonlinear interactions is studied by the dynamic renormalization group method. The long-range effect produces new fixed points with continuously varying exponents and gives distinct phase transitions, depending on both the long-range interaction strength and the substrate dimension $d$. The long-range interaction makes the surface width less rough than that of the short-range interaction. In particular, the surface becomes a smooth one with a negative roughness exponent at the physical dimension d=2.Comment: 4 pages(LaTex), 1 figure(Postscript