# Tropical curves and sandpiles

You can see our two talks on the BIRS-CMO conference Sandpiles Groups in Oaxaca 2015,

http://www.birs.ca/events/2015/5-day-workshops/15w5119/videos/watch/201511181000-KALININ.html

http://www.birs.ca/events/2015/5-day-workshops/15w5119/videos/watch/201511181100-Shkolnikov.html

Our presentations: oaxaca.pdf oaxacanovember15_proofs.pdf

See the simulation on Youtube: https://www.youtube.com/watch?v=7hC-vUuhb7c

**TROPICAL ANALYTIC CURVES IN 2-DIMENSIONAL SANDPILE MODEL, Nikita Kalinin, Mikhail Shkolnikov.**

Abstract: We consider the sandpile model on a convex part of the square lattice. 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 techniques to deal with particular integer-valued solutions of a certain Dirichlet problems. The continuous version of the considered relaxation reveals an interesting dynamics on polytopes.

Keywords:Tropical curves, sandpile model, tropical dynamics, discrete harmonic functions, Steiner problem, tropical symplectic area.

https://arxiv.org/abs/1509.02303

https://arxiv.org/abs/1502.06284

, and in the limit we always obtain something like :

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

# Code

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. Open it in “Spyder-app” which is provided within Anaconda.

3. On the top part of the file you can change some parametres, choose the boudary etc. But you can start by pressing F5 (run file), 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.