MARK KAC SEMINAR

A system of stochastic differential delay equations models the motion of a light-sensitive robot, exploring an inhomogeneous landscape. It can be approximated by a stochastic differential equation without delay and, in this form, studied using a multiscale expansion. The resulting dynamics depends on the value of the delay parameter, which can be positive or negative -- I will explain what this means and how it is realized experimentally. The results for a single robot are applied to predict an aggregation-deaggregation transition in many-robot systems, in a very good agreement with experiments. A nontechnical summary can be found at https://physics.aps.org/articles/v9/13.

IDLA is a growth model where a random aggregate is built recursively. At each step, a new particle is added inside the aggregate and performs a random walk until it exits it, adding this exit point to the aggregate. In the classical model, all particles are started at the origin. In Uniform IDLA, a new layer of self-dependence is added by having new particles start uniformly at random on the aggregate.

We will present some important results on the classical model, as well as its connection to the divisible sandpile model, which will help us in tackling the uniform starting point case. We will show that the normalized uniform IDLA aggregate, like in the classical case, admits the Euclidean ball as a limiting shape. The proof relies on gradually improving a priori bounds via coupling, as well as an alternative construction in which a genealogical tree of the particles is introduced.