Tropical curves in sandpiles

Abstract

We study a sandpile model on the set of the lattice points in a large lattice polygon. A small perturbation ψ of the maximal stable state μ≡3 is obtained by adding extra grains at several points. It appears that the result ψ∘ of the relaxation of ψ coincides with μ almost everywhere; the set where ψ∘≠μ is called the deviation locus. The scaling limit of the deviation locus turns out to be a distinguished tropical curve passing through the perturbation points.