arXiv subjects
Publications and source records attributed to Ellak Somfai.
Diffusion-limited aggregation is consistent with simple scaling. However, strong subdominant terms are present, and these can account for various earlier claims of anomalous scaling. We show this in detail for the case of multiscaling.
Off-lattice 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.
The effect of boundaries on the force distributions in granular media is illustrated by simulations of 2D packings of frictionless, Hertzian spheres. To elucidate discrepancies between experimental observations and theoretical predictions, we distinguish between the weight distribution {\cal P} (w) measured in experiments and analyzed in the q-model, and the distribution of interparticle forces P(f). The latter one is robust, while {\cal P}(w) can be obtained once the local packing geometry and P(f) are known. By manipulating the (boundary) geometry, we show that {\cal P}(w) can be varied drastically.
The creation of fractal clusters by diffusion limited aggregation (DLA) is studied by using iterated stochastic conformal maps following the method proposed recently by Hastings and Levitov. The object of interest is the function $Φ^{(n)}$ which conformally maps the exterior of the unit circle to the exterior of an $n$-particle DLA. The map $Φ^{(n)}$ is obtained from $n$ stochastic iterations of a function $ϕ$ that maps the unit circle to the unit circle with a bump. The scaling properties usually studied in the literature on DLA appear in a new light using this language. The dimension of the cluster is determined by the linear coefficient in the Laurent expansion of $Φ^{(n)}$, which asymptotically becomes a deterministic function of $n$. We find new relationships between the generalized dimensions of the harmonic measure and the scaling behavior of the Laurent coefficients.