Sandpile and tropical sandpile models

YMSC Probability Seminar

Organizers：

吴昊，杨帆，姜建平，顾陈琳，李文博

Speaker：

Nikita Kalinin (Guangdong Technion-Israel Institute of Technology)

Time：

Thur., 16:00-17:00, Jan. 15, 2026

Venue：

C548, Shuangqing Complex Building A

Title：

Sandpile and tropical sandpile models

Abstract:

The abelian sandpile is a cellular automaton on a finite graph, e.g. a finite subset of the planar lattice: sites with at least four grains “topple,” sending one grain to each neighbor, and the final relaxed configuration is independent of the toppling order. When grains are added repeatedly at random sites, the resulting avalanches (the set of sites that topple) exhibit scale invariance and power-law statistics; moreover, recurrent configurations are tightly linked to spanning trees and random spanning forests.

I will focus on a geometric scaling-limit regime: start from the maximal stable state on a large lattice polygon and add a small number of extra grains at a set P of points. I will explain why the deviation locus — where the relaxed state differs from the initial state — after rescaling converges to a distinguished tropical curve passing through P.

Building on this, we propose a continuous tropical dynamical model (a scaling limit of sandpiles) that still exhibits self-organized criticality, with the sandpile toppling function becoming piecewise linear and the emergent pattern is captured by the corner locus of an associated tropical series.