Abstract
Sandpile models exhibit fascinating pattern structures: patches, characterized by quadratic functions, and line-shaped patterns (also called solitons, webs, or linear defects). It was predicted by Dhar and Sadhu that sandpile patterns with line-like features may be described in terms of tropical geometry. We explain the main ideas and technical tools—tropical geometry and discrete superharmonic functions—used to rigorously establish certain properties of these patterns. It seems that the aforementioned tools have great potential for generalization and application in a variety of situations.
Highlights
Reviewed by: Deepak Dhar, Indian Institute of Science Education and Research, Pune, India Tridib Sadhu, Tata Institute of Fundamental Research, India
In this article we focus on so-called sandpile models, and firstly, discuss in section 2 how the patterns in that model were obtained in experimental computer physics, and secondly, we survey the main ideas permitting to study these patterns with mathematical rigor: discrete harmonic analysis, tropical geometry, toppling function, and the most technical part of the proofs, the lower bound
We use lemmata about superharmonic functions: if it would be possible to perform smoothings an infinite number of times, Lemma 1 would imply that Fk is linear with integer slope in a compact neighborhood of l, there exists a linear function with integer slope which is less than Fonly on a finite neighborhood of the corner locus
Summary
Animals show beautiful skin and wing patterns. Explaining how these come about has been a longstanding puzzle. In line with the Darwinian paradigm, an evolutionary biologist may suggest that formations of patterns on the skin of animals are visual traces of certain biological mechanisms that help survival in terms of natural selection In his seminal book, Thomson [1] argues that the geometry of patterns may be mainly dictated by chemical forces, albeit it is known that patterns may benefit their owners in certain cases. Thomson [1] argues that the geometry of patterns may be mainly dictated by chemical forces, albeit it is known that patterns may benefit their owners in certain cases In his famous paper on morphogenesis [2], Turing speculated on the mechanism behind pattern formation on the skin of animals and proposed the famous reaction diffusion system, which consists of an inhibitor and an activator with different diffusion rates. We mention open problems and new research directions when appropriate
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