Persistence of screening and self-criticality in the scale invariant dynamics of diffusion limited aggregation

Off-lattice noise reduced diffusion-limited aggregation in three dimensions

Using off-lattice noise reduction, it is possible to estimate the asymptotic properties of diffusion-limited aggregation clusters grown in three dimensions with greater accuracy than would otherwise be possible. The fractal dimension of these aggregates is found to be 2.50+/-0.01 , in agreement with earlier studies, and the asymptotic value of the relative penetration depth is xi/ R(dep) =0.122+/-0.002 . The multipole powers of the growth measure also exhibit universal asymptotes. The fixed point noise reduction is estimated to be epsilon(f) approximately 0.0035 , meaning that large clusters can be identified with a low noise regime. The slowest correction to scaling exponents are measured for a number of properties of the clusters, and the exponent for the relative penetration depth and quadrupole moment are found to be significantly different from each other. The relative penetration depth exhibits the slowest correction to scaling of all quantities, which is consistent with a theoretical result derived in two dimensions. We also note fast corrections to scaling, whose limited relevance is consistent with the requirement that clusters grow far enough in radius to support sufficient scales of ramification.

Diffusion-controlled growth: theory and closure approximations

We expand upon a new theoretical framework for diffusion-limited aggregation and associated dielectric breakdown models in two dimensions [R. C. Ball and E. Somfai, Phys. Rev. Lett. 89, 135503 (2002)]. Key steps are understanding how these models interrelate when the ultraviolet cut-off strategy is changed, the analogy with turbulence, and the use of logarithmic field variables. Within the simplest, Gaussian, truncation of mode-mode coupling, all properties can be calculated. The agreement with prior knowledge from simulations is encouraging, and a new superuniversality of the tip scaling exponent is discussed. We find angular resonances relatable to the cone angle theory, and we are led to predict a new screening transition in the DBM at large eta.

Off-lattice noise reduction and the ultimate scaling of diffusion-limited aggregation in two dimensions

Off-lattice diffusion-limited aggregation (DLA) clusters grown with different levels of noise reduction are found to be consistent with a simple fractal fixed point. Cluster shapes and their ensemble variation exhibit a dominant slowest correction to scaling, and this also accounts for the apparent "multiscaling" in the DLA mass distribution. We interpret the correction to scaling in terms of renormalized noise. The limiting value of this variable is strikingly small and is dominated by fluctuations in cluster shape. Earlier claims of anomalous scaling in DLA were misled by the slow approach to this small fixed point value.

Laplacian growth with separately controlled noise and anisotropy

Conformal mapping models are used to study the competition of noise and anisotropy in Laplacian growth. For this purpose, a family of models is introduced with the noise level and directional anisotropy controlled independently. Fractalization is observed in both anisotropic growth and growth with varying noise. The fractal dimension is determined from the cluster size scaling with cluster area. For isotropic growth d=1.7, at both high and low noise. For anisotropic growth with reduced noise the dimension can be as low as d=1.5 and apparently is not universal. Also, we study the fluctuations of particle areas and observe, in agreement with previous studies, that exceptionally large particles may appear during growth, leading to pathologically irregular clusters. This difficulty is circumvented by using an acceptance window for particle areas.

Diffusion-limited aggregation as a Markovian process: site-sticking conditions

Cylindrical lattice diffusion-limited aggregation, with a narrow width N, is solved for site-sticking conditions using a Markovian matrix method (which was previously developed for the bond-sticking case). This matrix contains the probabilities that the front moves from one configuration to another at each growth step, calculated exactly by solving the Laplace equation and using the proper normalization. The method is applied for a series of approximations, which include only a finite number of rows near the front. The fractal dimensionality of the aggregate is extrapolated to a value near 1.68.

Diffusion-limited aggregation as a markovian process: bond-sticking conditions

Cylindrical lattice diffusion limited aggregation (DLA), with a narrow width N, is solved using a Markovian matrix method. This matrix contains the probabilities that the front moves from one configuration to another at each growth step, calculated exactly by solving the Laplace equation and using the proper normalization. The method is applied for a series of approximations, which include only a finite number of rows near the front. The matrix is then used to find the weights of the steady-state growing configurations and the rate of approaching this steady-state stage. The former are then used to find the average upward growth probability, the average steady-state density and the fractal dimensionality of the aggregate, which is extrapolated to a value near 1.64.

Diffusion-limited aggregation: a revised mean-field approach

We propose a revision of the classic mean-field approach of diffusion-limited aggregation (DLA) model originally introduced by Witten and Sander [Phys. Rev. Lett. 47, 1400 (1981)]. The derived nonlinear mean-field equations providing lattice anisotropy are used to model diffusional growth on square lattice in linear and circular source geometries. The overall cluster shapes obtained from the mean-field calculations are found to satisfy the known scaling behavior experimentally observed for DLA simulations.