Self-organized criticality and pattern emergence through the lens of tropical geometry Journal Article


Author(s): Kalinin, Nikita; Guzmán-Sáenz, Aldo; Prieto Y; Shkolnikov, Mikhail; Kalinina V; Lupercio, Ernesto
Article Title: Self-organized criticality and pattern emergence through the lens of tropical geometry
Affiliation IST Austria
Abstract: Tropical geometry, an established field in pure mathematics, is a place where string theory, mirror symmetry, computational algebra, auction theory, and so forth meet and influence one another. 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.
Keywords: Tropical geometry; Power laws; Proportional growth; Self-organized criticality; pattern information
Journal Title: PNAS: Proceedings of the National Academy of Sciences of the United States of America
Volume: 115
Issue 35
ISSN: 00278424
Publisher: National Academy of Sciences  
Date Published: 2018-08-28
Start Page: E8135
End Page: E8142
DOI: 10.1073/pnas.1805847115
Notes: N.K. was funded by Swiss National Science Foundation PostDoc.Mobility Grant 168647, supported in part by a Young Russian Mathematics award, and Grant FORDECYT-265667 “Programa para un Avance Global e Integrado de la Matemática Mexicana.” Also, support from the Basic Research Program of the National Research University Higher School of Economics is gratefully acknowledged. Y.P. was funded by Grant FORDECYT-265667 and by “ABACUS” Laboratorio de Matemática Aplicada y Cómputo de Alto Rendimiento CINVESTAV-EDOMEX Proyecto CONACYT- EDOMEX-2011-01-165873 (Cinvestav). M.S. was supported by an Institute of Science and Technology Postdoctoral Fellowship program. E.L. thanks the Moshinsky Foundation, Conacyt, FORDECYT-265667, “ABACUS” Laboratorio de Matemática Aplicada y Cómputo de Alto Rendimiento CINVESTAV-EDOMEX Proyecto CONACYT- EDOMEX-2011-01-165873, Xiuhcoatl, Instituto de Matemáticas de la Universidad Nacional Autonoma de México, Samuel Gitler International Collaboration Center and the Laboratory of Mirror Symmetry National Research University Higher School of Economics, Russian Federation Government Grant 14.641.31.0001, and the kind hospitality of the University of Geneva and of the Mathematisches Forschungsinstitut Oberwolfach, where this work started.
Open access: no