TROPICAL CURVES IN 2-DIMENSIONAL SANDPILE MODEL
Nikita Sergeevich Kalinin, Mikhail Shkolnikov · arXiv (Cornell University) · 2015
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.