Pattern Formation and Tropical Geometry - Kalinin, Nikita (Frontiers in Physics, 2020)
Sandpile model exhibits fascinating pattern structures: patches, characterized by quadratic functions, and line-shaped patterns (also called solitons, webs, or linear defects). It was predicted by Dhar and Sadhu that sandpile patterns with line-like features may be described in terms of tropical geometry. We explain the main ideas and technical tools – tropical geometry and discrete superharmonic functions – used to rigorously establish certain properties of these patterns. It seems that the aforementioned tools have great potential for generalizing and applying in a variety of situations.