Universality in boundary domain growth by sudden bridging

We report on universality in boundary domain growth in cluster aggregation in the limit of maximum concentration. Maximal concentration means that the diffusivity of the clusters is effectively zero and, instead, clusters merge successively in a percolation process, which leads to a sudden growth of the boundary domains. For two-dimensional square lattices of linear dimension L, independent of the models studied here, we find that the maximum of the boundary interface width, the susceptibility χ, exhibits the scaling χ ~ L^γ with the universal exponent γ = 1. The rapid growth of the boundary domain at the percolation threshold, which is guaranteed to occur for almost any cluster percolation process, underlies the the universal scaling of χ.

Attracted diffusion-limited aggregation

In this paper we present results of extensive Monte Carlo simulations of diffusion-limited aggregation (DLA) with a seed placed on an attractive plane as a simple model in connection with the electrical double layers. We compute the fractal dimension of the aggregated patterns as a function of the attraction strength α. For the patterns grown in both two and three dimensions, the fractal dimension shows a significant dependence on the attraction strength for small values of α and approaches that of the ordinary two-dimensional (2D) DLA in the limit of large α. For the nonattracting case with α = 1, our results in three dimensions reproduce the patterns of 3D ordinary DLA, while in two dimensions our model leads to the formation of a compact cluster with dimension 2. For intermediate α, the 3D clusters have a quasi-2D structure with a fractal dimension very close to that of the ordinary 2D DLA. This allows one to control the morphology of a growing cluster by tuning a single external parameter α.

Classification of (2+1) -dimensional growing surfaces using Schramm-Loewner evolution

Statistical behavior and scaling properties of isoheight lines in three different saturated two-dimensional grown surfaces with controversial universality classes are investigated using ideas from Schramm-Loewner evolution (SLE_{κ}). We present some evidence that the isoheight lines in the ballistic deposition (BD), Eden and restricted solid-on-solid (RSOS) models have conformally invariant properties all in the same universality class as the self-avoiding random walk (SAW), equivalently SLE_{8/3}. This leads to the conclusion that all these discrete growth models fall into the same universality class as the Kardar-Parisi-Zhang (KPZ) equation in two dimensions.

Scaling and multiscaling behavior of the perimeter of a diffusion-limited aggregation generated by the Hastings-Levitov method

In this paper, we analyze the scaling behavior of a diffusion-limited aggregation (DLA) simulated by the Hastings-Levitov method. We obtain the fractal dimension of the clusters by direct analysis of the geometrical patterns, in good agreement with one obtained from an analytical approach. We compute the two-point density correlation function and we show that, in the large-size limit, it agrees with the obtained fractal dimension. These support the statistical agreement between the patterns and DLA clusters. We also investigate the scaling properties of various length scales and their fluctuations, related to the boundary of the cluster. We find that all of the length scales do not have a simple scaling with the same correction to scaling exponent. The fractal dimension of the perimeter is obtained equal to that of the cluster. The growth exponent is computed from the evolution of the interface width equal to β = 0.557(2). We also show that the perimeter of the DLA cluster has an asymptotic multiscaling behavior.

Scaling of clusters and winding-angle statistics of isoheight lines in two-dimensional Kardar-Parisi-Zhang surfaces

We investigate the statistics of isoheight lines of (2+1) -dimensional Kardar-Parisi-Zhang model at different level sets around the mean height in the saturation regime. We find that the exponent describing the distribution of the height-cluster size behaves differently for level cuts above and below the mean height, while the fractal dimensions of the height-clusters and their perimeters remain unchanged. The statistics of the winding angle confirms the previous observation that these contour lines are in the same universality class as self-avoiding random walks.

Direct evidence for conformal invariance of avalanche frontiers in sandpile models

Appreciation of stochastic Loewner evolution (SLE_{kappa}) , as a powerful tool to check for conformal invariant properties of geometrical features of critical systems has been rising. In this paper we use this method to check conformal invariance in sandpile models. Avalanche frontiers in Abelian sandpile model are numerically shown to be conformally invariant and can be described by SLE with diffusivity kappa=2 . This value is the same as value obtained for loop-erased random walks. The fractal dimension and Schramm's formula for left passage probability also suggest the same result. We also check the same properties for Zhang's sandpile model.

Conformal invariance of isoheight lines in a two-dimensional Kardar-Parisi-Zhang surface

The statistics of isoheight lines in the (2+1) -dimensional Kardar-Parisi-Zhang (KPZ) model is shown to be conformally invariant and equivalent to those of self-avoiding random walks. This leads to a rich variety of exact analytical results for the KPZ dynamics. We present direct evidence that the isoheight lines can be described by the family of conformally invariant curves called Schramm-Loewner evolution (or SLE_{kappa} ) with diffusivity kappa=8/3 . It is shown that the absence of the nonlinear term in the KPZ equation will change the diffusivity kappa from 8/3 to 4, indicating that the isoheight lines of the Edwards-Wilkinson surface are also conformally invariant and belong to the universality class of domain walls in the O(2) spin model.

Conformal curves on the WO3 surface

We have studied the isoheight lines on the WO3 surface as a physical candidate for conformally invariant curves. We have shown that these lines are conformally invariant with the same statistics of domain walls in the critical Ising model. They belong to the family of conformal invariant curves called Schramm-Loewner evolution (or SLE(kappa)), with diffusivity of kappa approximately 3. This can be regarded as the first experimental observation of SLE curves. We have also argued that Ballistic Deposition (BD) can serve as a growth model giving rise to contours with similar statistics at large scales.