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.

sand.pdf

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.

sandnew.pdf

and in the limit we always obtain something like

Soon, we will upload here code and more pictures.

For more information about sand read http://people.reed.edu/~davidp/sand/lit/literature.html and http://nautil.us/issue/23/dominoes/the-amazing-autotuning-sandpile

1. Install Anaconda http://continuum.io/downloads

or manage to use the package Tkinter by yourself.

2. Use the file sandsimulation.pdf, change its extension to .py, i.e. sandsimulation.py.

3. On the top part of the file you can change some parametres, choose the boudary etc. But you can start by clicking and observing what happens.

4. Basically, we start from the grid with 3 grains of sand everywhere. When you left-click on the left half of the screen you add an additional grain of sand. White color means that at this point we have 3 grains of sand.

5. You can rescale the picture if you left-click in the top half of the right half of the screen. You can undo return to the previous picture by clicking in the bottom half of the right half of the screen.