The critical 1D multi-particle DLA.
In multi-particle Diffusion Limited Aggregation (DLA) a sea of particles performs independent random walks until they run into the aggregate and are absorbed. In dimension 1, the rate of growth of the aggregate depends on $\lambda$, the density of the particles. Kesten and Sidoravicius proved that when $\lambda < 1$ the aggregate grows like $t^{1/2}$. They furthermore predicted linear growth when $\lambda > 1$ (subsequently confirmed by Sly) and $t^{2/3}$ growth at the critical density $\lambda = 1$. We address the critical case, confirming the $t^{2/3}$ growth and show that aggregate has a scaling limit whose derivative is a self-similar diffusion. Surprisingly, this contradicts conjectures on the speed in the mildly supercritical regime when $\lambda = 1 + \epsilon$.
Joint work with Dor Elboim and Allan Sly