Limit points of external DLA models in dimension two

Speaker:

X. Zhang, Department of Applied Mathematics, Illinois Institute of Technology

Description:

Diffusion-limited aggregations (DLA) are random growth models on the lattice, where a set grows at the boundary sites when triggered by the absorption of a sequence of lattice-valued stochastic process (particles); a typical example is the self-avoiding random walk. We are interested in the limiting behavior of such sets as the grid size decreases, with the empirical law of the underlying particles converging to the Wiener measure. The main discovery of this work is that almost surely the planar Wiener process makes a loop around itself infinitely many times. Based on this geometric observation we show that the limit points of a certain class of external DLA models coincide a Wiener process stopped upon hitting the limiting set. Furthermore I will talk about the connection between this problem and the supercooled Steffan problem. In particular we will see that the latter in two dimensions cannot be approximated by external DLA models.

This one-hour talk will be followed by a 15-minute question and answer session.

This is a joint work with Sergey Nadtochiy and Mykhaylo Shkolnikov.

 

Mathematical Finance, Stochastic Analysis, and Machine Learning