Recent work

Harmonic activation and transport, or HAT, is a Markov chain with the following dynamics. Start with a configuration—a finite subset $U$ of $\mathbb{Z}^2$ with at least two elements. Form the next configuration as $U\cup\{y\}{\setminus}\{x\}$, where $x$ is chosen according to the harmonic measure of $U$, and where $y$ is the place a random walk from $x$ leaves from when it first enters $U{\setminus}\{x\}$. We say that $x$ is "activated" and "transported" to $y$. HAT is the process which results from iterating this procedure.

In this paper, we proved that HAT exhibits a phenomenon called collapse, wherein the diameter of a configuration is reduced to its logarithm, over a number of steps proportional to this logarithm. Collapse implies the existence of the stationary distribution of HAT (supported on configurations which are defined up to translation) and the exponential tightness of diameter under it. It also produces a renewal structure which we used to prove that the center of mass process, properly rescaled, converges in distribution to two-dimensional Brownian motion.

We also proved new estimates of harmonic measure which have consequences for Laplacian growth models like diffusion-limited aggregation.

In his article , Nigel Franks wrote "[I]f 100 army ants are placed on a flat surface, they will walk around and around. . .until they die of exhaustion. In extremely high numbers, however, it is a different story." If you could add ants, one at a time, how many would you need to add before the "different story" emerged?

The main result of this paper is that HAT exhibits a remarkable phase transition, wherein the addition of a single particle or dimension changes the limiting behavior from one of stationarity to one of transience. Furthermore, transience occurs in only one "way": The group of particles splits into clusters of two or three particles, but no other number, which become increasingly separated.

In this analogy: army ants = particles; flat surface = low dimension; walking around and around = stationarity; different story = clustering and transience.