tropicalsand

# Tropical curves and sandpiles

then and, finally,

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

Our presentations: oaxaca.pdf oaxacanovember15_proofs.pdf

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.

, and in the limit we always obtain something like :