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Scaling and Crossovers in Diffusion Limited Aggregation

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 Added by Leonard M. Sander
 Publication date 1999
  fields Physics
and research's language is English




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We discuss the scaling of characteristic lengths in diffusion limited aggregation (DLA) clusters in light of recent developments using conformal maps. We are led to the conjecture that the apparently anomalous scaling of lengths is due to one slow crossover. This is supported by an analytical argument for the scaling of the penetration depth of newly arrived random walkers, and by numerical evidence on the Laurent coefficients which uniquely determine each cluster. We find a single crossover exponent of -0.3 for all the characteristic lengths in DLA. This gives a hint about the structure of the renormalization group for this problem.



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185 - Ellak Somfai 2003
We performed extensive numerical simulation of diffusion-limited aggregation in two dimensional channel geometry. Contrary to earlier claims, the measured fractal dimension D = 1.712 +- 0.002 and its leading correction to scaling are the same as in the radial case. The average cluster, defined as the average conformal map, is similar but not identical to Saffman-Taylor fingers.
We investigate diffusion-limited aggregation (DLA) in a wedge geometry. Arneodo and collaborators have suggested that the ensemble average of DLA cluster density should be close to the noise-free selected Saffman-Taylor finger. We show that a different, but related, ensemble average, that of the conformal maps associated with random clusters, yields a non-trivial shape which is also not far from the Saffman-Taylor finger. However, we have previously demonstrated that the same average of DLA in a channel geometry is not the Saffman-Taylor finger. This casts doubt on the idea that the average of noisy diffusion-limited growth is governed by a simple transcription of noise-free results.
164 - Patrick B. Warren 2008
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101 - Mogens H. Jensen 2001
The method of iterated conformal maps allows to study the harmonic measure of Diffusion Limited Aggregates with unprecedented accuracy. We employ this method to explore the multifractal properties of the measure, including the scaling of the measure in the deepest fjords that were hitherto screened away from any numerical probing. We resolve probabilities as small as $10^{-35}$, and present an accurate determination of the generalized dimensions and the spectrum of singularities. We show that the generalized dimensions $D_q$ are infinite for $q<q^*$, where $q^*$ is of the order of -0.2. In the language of $f(alpha)$ this means that $alpha_{max}$ is finite. The $f(alpha)$ curve loses analyticity (the phenomenon of phase transition) at $alpha_{max}$ and a finite value of $f(alpha_{max})$. We consider the geometric structure of the regions that support the lowest parts of the harmonic measure, and thus offer an explanation for the phase transition, rationalizing the value of $q^*$ and $f(alpha_{max})$. We thus offer a satisfactory physical picture of the scaling properties of this multifractal measure.
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