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Scaling exponent of the maximum growth probability in diffusion-limited aggregation
Mogens H Jensen1, Joachim Mathiesen, Itamar Procaccia
1The Niels Bohr Institute, Blegdamsvej 17, Copenhagen, Denmark.
Summary
The Turkevich-Scher conjecture in multifractal theory for diffusion limited aggregation (DLA) has been disproven. New measurements show the fractal dimension (D) and alpha(min) exponent do not fit the proposed D=1+alpha(min) relation.
Area of Science:
- Complex Systems
- Statistical Physics
- Fractal Geometry
Background:
- The Turkevich-Scher conjecture proposed a relationship between the fractal dimension (D) and the alpha(min) exponent in diffusion limited aggregation (DLA).
- This conjecture, D=1+alpha(min), remained untested due to challenges in accurately measuring both D and alpha(min).
Purpose of the Study:
- To accurately determine the alpha(min) exponent for diffusion limited aggregation (DLA).
- To rigorously test the validity of the Turkevich-Scher conjecture.
Main Methods:
- Utilized the method of iterated conformal maps to precisely calculate the fractal dimension (D).
- Accurately measured the alpha(min) exponent, which characterizes the hottest regions of the harmonic measure.
Main Results:
- The fractal dimension (D) was determined to be 1.713 ± 0.003.
- The alpha(min) exponent was found to be 0.665 ± 0.004.
Conclusions:
- The measured values of D and alpha(min) do not satisfy the Turkevich-Scher conjecture (D = 1 + alpha(min)).
- The study concludes that the Turkevich-Scher conjecture is incorrect for diffusion limited aggregation (DLA).