Diffusion-limited Aggregation and Multifractality: A Historical Perspective
Mardi 12/12/2023, 11:00-12:00
Amphi Claude Bloch, Bât. 774, Orme des Merisiers
A little over forty years ago, a simple algorithm, “Diffusion-limited aggregation” (DLA), was introduced by Witten and Sander as a model for solidification in colloids. It quickly became evident, both from theoretical analyses, and from empirical observation, that this model was a useful model to describe pattern formation in a wide variety of settings— colloidal aggregation, two-phase flow, formation of solid dendrites, and a variety of other problems. All of these systems develop branched growth features that are distinctive and highly recognizable.
The standard theoretical formalism for understanding these patterns, multifractal analysis, emerged in tandem— the measure displaying the spectrum of fractal behaviors being the growth, or harmonic, measure of the surface. Our knowledge of scaling behaviors of diffusion-limited aggregates is limited to relationships among these multifractal exponents, not only for DLA, but also for the generalized “Dielectric Breakdown Models”, which include the famous Eden model as a limit, and DLA as a special case. A unified approach to these models in dimensionality d=2 was advanced by Hastings and Levitov based on iterated conformal maps, which has proven to be a powerful technique, but has not yet allowed direct computation of scaling properties, except in special cases. While the 70s and 80s were notable for the number of equilibrium and non-equilibrium statistical physics problems that were largely solved by then-existing theoretical methods, DLA remains an exception.
I will review this history, and the features of DLA that make it both fascinating and still challenging. In the last decade, links between DLA and the Schramm-Loewner evolution (SLE) and the emergence of solutions in Liouville quantum gravity (LQG) have only deepened the mystery of DLA.