Self-Organized Criticality and Pattern Emergence through the lens of Tropical Geometry

Abstract

Tropical Geometry, an established field in pure mathematics, is a place where String Theory, Mirror Symmetry, Computational Algebra, Auction Theory, etc, meet and influence each other. In this paper, we report on our discovery of a tropical model with self-organized criticality (SOC) behavior. Our model is continuous, in contrast to all known models of SOC, and is a certain scaling limit of the sandpile model, the first and archetypical model of SOC. We describe how our model is related to pattern formation and proportional growth phenomena, and discuss the dichotomy between continuous and discrete models in several contexts. Our aim in this context is to present an idealized tropical toy-model (cf. Turing reaction-diffusion model), requiring further investigation.

32 pages, 7 Figures

References in corpus (3)

• Patchworking real algebraic varieties
• Lagrangian pairs of pants
• Conservation laws for strings in the Abelian Sandpile Model

Cited by in corpus (10)

• Network Geometry
• Multi-scale structural complexity of natural patterns
• Harmonic dynamics of the Abelian sandpile
• Fundamental interactions in self-organized critical dynamics on higher-order networks
• A Tropical Geometric Approach To Exceptional Points
• Spatiotemporal fluctuation scaling law and metapopulation modeling of the novel coronavirus (COVID-19) and SARS outbreaks
• Fragmented perspective of self-organized criticality and disorder in log gravity
• Relaxation in one-dimensional tropical sandpile
• Some statistics about Tropical Sandpile Model
• Emergence of power-law distributions in self-segregation reaction-diffusion processes