Introduction to tropical series and wave dynamic on them

Kalinin, Nikita; Shkolnikov, Mikhail

doi:10.48550/arxiv.1706.03062

Cited by 4 publications

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“…This equality was observed earlier in [2]. Note also that p 2 + q 2 is the symplectic area of an edge (p, q) in a tropical curve (see [11] for details).…”

Section: Proof Of Theorem 2 For ψ Edgesupporting

confidence: 73%

“…Let F be any function A → Z. Lemma 6.3. (Used on pages [10,11,13]) In the above hypothesis the following equality holds:…”

Section: Discrete Superharmonic Integer-valued Functionsmentioning

confidence: 99%

“…In addition, we accurately write the theory of sandpiles on infinite domains in the absence of references, though it is absolutely parallel to the finite case. This article contains the facts (Theorem 2, Corollary 2.10) that we need later to establish more general convergence in sandpiles, see [12], from where we have extracted for better readability [11] and this article.…”

Section: Introductionmentioning

confidence: 99%

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Sandpile solitons via smoothing of superharmonic functions

Kalinin¹,

Shkolnikov²

2017

Preprint

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Let F : Z 2 → Z be the pointwise minimum of several linear functions. The theory of smoothing of integer-valued superharmonic function allows us to prove that under certain conditions there exists the pointwise minimal superharmonic function which coincides with F "at infinity".We develop such a theory to prove existence of so-called solitons (or strings) in a certain sandpile model, studied by S. Caracciolo, G. Paoletti, and A. Sportiello. Thus we made a step towards understanding the phenomena of the identity in the sandpile group for a square where solitons appear according to experiments. We prove that sandpile states, defined using our smoothing procedure, move changeless when we send waves (that is why we call them solitons), and can interact, forming triads and nodes.

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“…This equality was observed earlier in [2]. Note also that p 2 + q 2 is the symplectic area of an edge (p, q) in a tropical curve (see [11] for details).…”

Section: Proof Of Theorem 2 For ψ Edgesupporting

confidence: 73%

“…Let F be any function A → Z. Lemma 6.3. (Used on pages [10,11,13]) In the above hypothesis the following equality holds:…”

Section: Discrete Superharmonic Integer-valued Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Sandpile solitons via smoothing of superharmonic functions

Kalinin¹,

Shkolnikov²

2017

Preprint

Self Cite

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“…Theorem 2 in [6] is proven in Section 8. Theorem 3 in [6] easily follows from Theorem 1, and is proven in [4]. Theorem 4 in [6] follows from Theorem 1 here just because of the definition of the function f Ω,P (see Definition 1.5).…”

Section: Proof Of the Lower Estimatementioning

confidence: 85%

“…For each Ω-tropical series there exists a unique canonical form of it. As we proved in [4], for each Ω-tropical series f , for each z ∈ Ω • , f is given by the minimum of a finite number of monomials a ij + ix + jy in a small neighborhood of z. Therefore locally C(f ) is a graph with straight edges.…”

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confidence: 80%

Tropical curves in sandpiles

Kalinin

Shkolnikov

2016

Comptes Rendus. Mathématique

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Abstract. We study a sandpile model on the set of the lattice points in a large lattice polygon. A small perturbation ψ of the maximal stable state µ ≡ 3 is obtained by adding extra grains at several points. It appears, that the result ψ • of the relaxation of ψ coincides with µ almost everywhere; the set where ψ • = µ is called the deviation locus. The scaling limit of the deviation locus turns out to be a distinguished tropical curve passing through the perturbation points.Nous considérons le modèle du tas de sable sur l'ensemble des points entiers d'un polygone entier. En ajoutant des grains de sable en certains points, on obtient une perturbation mineure de la configuration stable maximale µ ≡ 3. Le résultat ψ • de la relaxation est presque partoutégalà µ. On appelle lieu de déviation l'ensemble des points où ψ • = µ. La limite au sens de la distance de Hausdorff du lieu de déviation est une courbe tropicale spéciale, qui passe par les points de perturbation.

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

Kalinin

Guzmán-Sáenz

Prieto

et al. 2018

Proc. Natl. Acad. Sci. U.S.A.

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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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Introduction to tropical series and wave dynamic on them

Cited by 4 publications

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Sandpile solitons via smoothing of superharmonic functions

Sandpile solitons via smoothing of superharmonic functions

Tropical curves in sandpiles

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

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