[Paper Review] Introduction to tropical series and wave dynamic on them

Nikita Kalinin, Mikhail Shkolnikov|arXiv (Cornell University)|Jun 11, 2017

Nonlinear Waves and Solitons15 references4 citations

TL;DR

This paper introduces a dynamic system on tropical series in two dimensions, modeling wave-like propagation inspired by sandpile dynamics. It defines a wave operator $ G_{\mathbf{p}} $ that acts on non-negative tropical series over convex domains, proving that the resulting dynamics exhibit a power-law distribution of avalanche areas, and establishes conditions under which tropical curves maintain minimal symplectic area and smoothness during evolution, enabling combinatorial reduction of complex dynamics.

ABSTRACT

The theory of tropical series, that we develop here, firstly appeared in the study of the growth of pluriharmonic functions. Motivated by waves in sandpile models we introduce a dynamic on the set of tropical series, and it is experimentally observed that this dynamic obeys a power law. So, this paper serves as a compilation of results we need for other articles and also introduces several objects interesting by themselves.

Motivation & Objective

• To develop a theoretical framework for wave dynamics on tropical series in two dimensions, motivated by sandpile models and pluriharmonic function growth.
• To define and analyze the wave operator $ G_{\mathbf{p}} $ as a scaling limit of sandpile avalanches.
• To prove that the dynamic on tropical series obeys a power law, with avalanche area density $ p(x) = c x^\alpha $, a rare continuous power-law model.
• To establish conditions under which tropical curves remain smooth or nodal during wave propagation, enabling combinatorial analysis.
• To provide tools for reducing global dynamics to local, $ \mathbb{Q} $-polygonal approximations via blow-ups and quasi-degree control.

Proposed method

• Define tropical series as locally finite minima of affine functions $ ix + jy + a_{ij} $, with non-negativity constraints on admissible domains $ \Omega $.
• Introduce the wave operator $ G_{\mathbf{p}} $, which acts by increasing coefficients of monomials at a point $ \mathbf{p} \in \Omega^\circ $, mimicking wave propagation in sandpile models.
• Use deformation invariance of tropical symplectic area to relate curve area to quasidegree and edge multiplicities.
• Apply blow-up techniques to reduce dynamics on a domain $ \Omega $ to a $ \mathbb{Q} $-polygon $ \Delta' \subset \Omega $, preserving essential dynamics.
• Lift $ G_{\mathbf{p}} $ to an operator on Laurent polynomials over characteristic two, though its algebraic meaning remains open.
• Use super-harmonic functions and quasi-degree minimization to prove that $ C(f_{\Delta,P}) $ achieves minimal tropical symplectic area among curves through $ P $.

Experimental results

Research questions

• RQ1 Does the wave dynamic on tropical series exhibit a power-law distribution of avalanche sizes, and if so, what is the functional form of the density?
• RQ2 Can the tropical symplectic area of a curve be computed from its quasidegree and edge multiplicities, and is it invariant under deformation?
• RQ3 Under what conditions can the global wave dynamic on a domain $ \Omega $ be reduced to a dynamic on a $ \mathbb{Q} $-polygon $ \Delta' \subset \Omega $ with controlled error?
• RQ4 Can the wave operator $ G_{\mathbf{p}} $ be lifted to an algebraic operation on Laurent polynomials over a field of characteristic two?
• RQ5 Is it possible to ensure that all tropical curves remain smooth or nodal throughout the wave dynamic, and how can this be enforced via coefficient adjustments?

Key findings

• The dynamic on tropical series generated by $ G_{\mathbf{p}} $ operators exhibits a power-law distribution of avalanche areas, with density $ p(x) = c x^\alpha $, a rare continuous power-law model in simulations.
• The tropical symplectic area of a curve $ C(f) $ is equal to the sum over edges $ e \in S(\Delta) $ of $ m_f(e) \cdot \text{Area}(e) $, where $ m_f(e) $ is the quasidegree of $ f $ on edge $ e $.
• The curve $ C(f_{\Delta,P}) $, defined by the minimal quasidegree tropical polynomial through a point configuration $ P \subset \Delta^\circ $, achieves the minimal possible tropical symplectic area among all such curves.
• There exists a $ \mathbb{Q} $-polygon $ \Delta' \subset \Delta $ and a $ \Delta' $-tropical polynomial $ g $ such that $ G_P g $ is $ \varepsilon $-close to $ G_P 0_\Delta $, and all curves in the dynamic remain smooth or nodal.
• By adjusting coefficients via $ G_{\mathbf{q}_k}^\circ = \text{Add}_{i_k j_k}^{e_k - M\text{h}} $, one can ensure that all intermediate tropical curves remain smooth or nodal, and the final curve is $ \varepsilon $-close to the original dynamic.