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

TL;DR

This paper introduces a novel continuous tropical geometry model exhibiting self-organized criticality, linking pattern formation and growth phenomena, and contrasting it with traditional discrete models like the sandpile.

Contribution

It presents the first continuous tropical model with SOC behavior, derived as a scaling limit of the sandpile, expanding the understanding of SOC in mathematical physics.

Findings

The model demonstrates SOC behavior in a continuous setting.

It relates tropical geometry to pattern formation and growth.

The model offers a new perspective contrasting discrete and continuous SOC models.

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.