Time: October 17, 2023, 2:00 pm

Place: Room 403, IMI – BAS

Speaker: Mikhail Shkolnikov, IMI-BAS

Title: Tropical structures in sandpile model

Abstract: The sandpile model is a cellular automaton that can be defined on any graph. Though it has appeared independently in a variety of contexts, it attracted tremendous attention from the wide scientific community when its instance on a large domain of the square lattice was put forward as the simplest rigorous example of self-organized criticality. Such phenomena, real-life examples of which are earthquakes, solar flares, or neuronal avalanches, could be vaguely described as being between order and chaos, demonstrating power laws without being finely tuned, and transcending the scale. Tropical geometry, on the other hand, is a field of pure mathematics often presented as a combinatorial version of algebraic geometry. I will tell how tropical curves arise in the scaling limit of the sandpile model in the vicinity of the maximal stable state and explain two major consequences inspired by this fact. The first one is that there is a continuous model for self-organized criticality, the only known model of a kind, defined in the realm of tropical geometry. The second is that the totality of recurrent states in the original sandpile model, the sandpile group, approximates a continuous group, a tropical Abelian variety, which is functorial with respect to inclusions of domains, allowing to compute its scaling limit as a space of circle-valued harmonic functions on the whole lattice.