  Vicsek fractal graphs are an important class of infinite graphs with self
similar properties, polynomial growth and treelike features, on which several
dynamical processes such as random walks or Abelian sandpiles can be rigorously
analyzed and one can obtain explicit closed form expressions. While such
processes on Vicsek fractals and on Euclidean lattices $\mathbb{Z}^2$ share
some properties for instance in the recurrence behaviour, many quantities
related to sandpiles on Euclidean lattices are still poorly understood. The
current work focuses on the stabilization/explosion of Abelian sandpiles on
Vicsek fractal graphs, and we prove that a sandpile sampled from the infinite
volume limit plus one additional particle stabilizes with probability 3/4, that
is, it does not stabilize almost surely and it explodes with the complementary
probability 1/4. We prove the main result by using two different approaches:
one of probabilistic nature and one of algebraic flavor. The first approach is
based on investigating the particles sent to the boundary of finite volumes and
showing that their number stays above four with positive probability. In the
second approach we relate the question of stabilization and explosion of
sandpiles in infinite volume to the order of elements of the sandpile group on
finite approximations of the infinite Vicsek graph. The method applies to more
general state spaces and by employing it we also find all invariant factors of
the sandpile groups on the finite approximations of the infinite Vicsek
fractal.
