  In the Diffusion Limited Aggregation (DLA) process on on $\mathbb{Z}^2$, or
more generally $\mathbb{Z}^d$, particles aggregate to an initially occupied
origin by arrivals on a random walk. The scaling limit of the result,
empirically, is a fractal with dimension strictly less than $d$. Very little
has been shown rigorously about the process, however.
  We study an analogous process on the Boolean lattice $\{0,1\}^n$, in which
particles take random decreasing walks from $(1,\dots,1)$, and stick at the
last vertex before they encounter an occupied site for the first time; the
vertex $(0,\dots,0)$ is initially occupied. In this model, we can rigorously
prove that lower levels of the lattice become full, and that the process ends
by producing an isolated path of unbounded length reaching $(1,\dots,1)$.
