Ceci est une ancienne révision du document !
Tropical curves and sandpiles
then 
and, finally, 
Tropical curves in $2$-dimensional sandpile model, Nikita Kalinin, Mikhail Shkolnikov.
Abstract: We consider the sandpile model on a part of the square lattice, bounded by a polygon. We modify the maximal stable state by adding a grain of sand at each of the $n$ fixed points: the consequent relaxation produces pictures where we can see tropical curves. These curves pass through the same $n$ fixed points and solve a version of the Steiner tree problem: minimization of {\it tropical symplectic area}. In order to show this, we develop several technics to deal with particular integer-valued solutions of certain Dirichlet problems and to study the continuous version of the considered relaxation which reveals an interesting dynamics on polytopes.
Keywords:Tropical curves, sandpile model, tropical dynamics, discrete harmonic functions, Steiner problem.
Tropical series and sandpiles in arbitrary convex domains, Nikita Kalinin, Mikhail Shkolnikov.
This is a draft of the second part of the article entitled ``Tropical curves in $2$-dimensional sandpile model''. Here we study the case of an arbitrary boundary.
and in the limit we always obtain something like
Soon, we will upload here code and more pictures.