Diffusion-limited aggregation is a stochastic process that grows a connected cluster by successively attaching particles arriving along
random walks. In the lattice model on 
, the cluster starts at the origin. Each new site is sampled
from the limiting first-contact distribution of a random
walk started far from the cluster and is added when the walk first reaches a
vacant site adjacent to the cluster. In transient dimensions 
, this first-contact distribution is conditioned on reaching
that set (Witten and Sander 1981, Dembo and Junge 2026).
Let 
be the cluster of 
sites. For a bounded open set 
of positive volume with boundary
of Lebesgue measure 0, put 
. Dembo and Junge (2026)
proved that, for each 
, there is 
such that, with probability
1,
 |
simultaneously for all such 
. Thus convergence in relative volume to a regular bounded
shape is impossible. They also proved 
, where 
consists of nearest-neighbor edges
with exactly one endpoint in 
. This boundary includes edges around internal holes.
Dembo and Junge (2026) report that ChatGPT generated the proofs under their direction.
