The Abelian Sandpile Model is a model for avalanches in
out-of-equilibrium statistical mechanics, paradigm of self-organised
criticality. In the 90's, mostly Dhar and collaborators elucidated a
series of remarkable exact results in this model, which related to
various branches of combinatorics. Among other things, a group
structure on the configurations of the system emerged.
The simplest geometry is a square portion of the 2D square lattice. In
this case, the group-identity configuration shows, in the
thermodynamic limit, convergence to a peculiar fractal shape, composed
of infinitely-many polygons, filled with different bi-periodic
patterns. The determination of this shape has been an open problem for
a few decades, and is solved here.
