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

N. Kalinin; A. Guzmán-Sáenz; Y. Prieto; M. Shkolnikov; V. Kalinina; E. Lupercio

doi:10.1073/pnas.1805847115

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

N. Kalinin, A. Guzmán-Sáenz, Y. Prieto, M. Shkolnikov, V. Kalinina, E. Lupercio^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

16 Scopus citations

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.

Original language	English
Pages (from-to)	E8135-E8142
Journal	Proceedings of the National Academy of Sciences of the United States of America
Volume	115
Issue number	35
DOIs	https://doi.org/10.1073/pnas.1805847115
State	Published - 2018
Externally published	Yes

Keywords

Pattern formation
Power laws
Proportional growth
Self-organized criticality
Tropical geometry

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10.1073/pnas.1805847115

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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.",

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AU - Kalinin, N.

AU - Guzmán-Sáenz, A.

AU - Prieto, Y.

AU - Shkolnikov, M.

AU - Kalinina, V.

AU - Lupercio, E.

PY - 2018

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AB - 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.

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

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Keywords

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