Multifractal Structure of the Harmonic Measure of Diffusion Limited Aggregates

@misc{citeulike:833430, abstract = {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.}, author = {Jensen, Mogens H. and Levermann, Anders and Mathiesen, Joachim and Procaccia, Itamar }, citeulike-article-id = {833430}, eprint = {cond-mat/0110203}, keywords = {dla, harmonic, multifractal}, month = {Oct}, posted-at = {2006-09-07 08:57:32}, priority = {2}, title = {Multifractal Structure of the Harmonic Measure of Diffusion Limited Aggregates}, url = {http://arxiv.org/abs/cond-mat/0110203}, year = {2001} }

TY - ELEC ID - citeulike:833430 L3 - citeulike-article-id:833430 TI - Multifractal Structure of the Harmonic Measure of Diffusion Limited Aggregates N2 - 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. KW - dla KW - harmonic KW - multifractal AU - Jensen, Mogens AU - Levermann, Anders AU - Mathiesen, Joachim AU - Procaccia, Itamar PY - 2001/Oct/10/ UR - http://arxiv.org/abs/cond-mat/0110203 ER -