24 citations · 108 across the 21 of their papers we have counts for
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The 2-Adic Valuation of the Order of the All-Ones Class in the Sandpile Group of a Square
Turgay Akyar, Artem Beliakov, Konstantin Delchev +6
Place one grain at every nonsink vertex of the wired square, and let be the order of this operation in the sandpile group. Thus is the least positive …
Relaxation in one-dimensional tropical sandpile
Mikhail Shkolnikov
A relaxation in the tropical sandpile model is a process of deforming a tropical hypersurface towards a finite collection of points. We show that, in the one-dimensional case, a re…
Sandpile solitons via smoothing of superharmonic functions
Nikita Kalinin, Mikhail Shkolnikov
Let be the pointwise minimum of several linear functions. The theory of smoothing of integer-valued superharmonic function allows us to prove that unde…
Introduction to tropical series and wave dynamic on them
Nikita Kalinin, Mikhail Shkolnikov
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 dy…
Tropical curves in sandpiles
Nikita Kalinin, Mikhail Shkolnikov
We study a sandpile model on the set of the lattice points in a large lattice polygon. A small perturbation of the maximal stable state is obtained by adding extra…
Sandpiles on the heptagonal tiling
Nikita Kalinin, Mikhail Shkolnikov
We study perturbations of the maximal stable state in a sandpile model on the set of faces of the heptagonal tiling on the hyperbolic plane. An explicit description for relaxations…
Tropical curves in sandpile models
Nikita Kalinin, Mikhail Shkolnikov
A sandpile is a cellular automaton on a graph that evolves by the following toppling rule: if the number of grains at a vertex is at least its valency, then this vertex sends one g…