Procaccia Eviatar B, Procaccia Itamar
Faculty of Industrial Engineering and Management, The Technion, Haifa 32000, Israel.
Department of Mathematics, Texas A&M University, College Station, Texas 77840, USA.
Phys Rev E. 2021 Feb;103(2):L020101. doi: 10.1103/PhysRevE.103.L020101.
Diffusion-limited aggregation (DLA) has served for 40 years as a paradigmatic example for the creation of fractal growth patterns. In spite of thousands of references, no exact result for the fractal dimension D of DLA is known. In this Letter we announce an exact result for off-lattice DLA grown on a line embedded in the plane D=3/2. The result relies on representing DLA with iterated conformal maps, allowing one to prove self-affinity, a proper scaling limit, and a well-defined fractal dimension. Mathematical proofs of the main results are available in N. Berger, E. B. Procaccia, and A. Turner, Growth of stationary Hastings-Levitov, arXiv:2008.05792.
扩散限制凝聚（DLA）四十年来一直是分形生长模式形成的典型例子。尽管有数千篇参考文献，但关于DLA的分形维数D的确切结果仍不为人知。在本快报中，我们宣布了在嵌入平面的直线上生长的非格点DLA的精确结果：D = 3/2。该结果依赖于用迭代共形映射来表示DLA，从而能够证明自相似性、恰当的标度极限以及明确定义的分形维数。主要结果的数学证明可在N. 伯杰、E. B. 普罗卡西亚和A. 特纳所著的《平稳黑斯廷斯 - 列维托夫的生长》（arXiv:2008.05792）中找到。
