Computer Science Laboratory of Paris-North University, University of Paris 13, Paris

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 recurrent 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. Here we describe the solution.