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Mikhail Shkolnikov

Publications and source records attributed to Mikhail Shkolnikov.

28 records · Page 2Linked to original sources
arXiv2019

Sandpile monomorphisms and limits

We introduce a tiling problem between bounded open convex polyforms $\hat{P}\subset\mathbb{R}^2$ with directed and uniquely colored edges. If there exists a tiling of the polyform $\hat{P}_2$ by $\hat{P}_1$, we show that one can construct a monomorphism from the sandpile group $G_{Γ_1}=\mathbb{Z}^{Γ_1}/Δ(\mathbb{Z}^{Γ_1})$ on the domain (graph) $Γ_1=\hat{P}_1\cap\mathbb{Z}^2$ to the respective group on $Γ_2=\hat{P}_2\cap\mathbb{Z}^2$. We provide several examples of infinite series of such tilings with polyforms converging to $\mathbb{R}^2$, and thus the first definition of scaling-limits for the sandpile group on the plane. Additional results include an exact sequence relating sandpile configurations to harmonic functions, an alternative formula for the order of the sandpile group based on a basis for the module of integer-valued harmonic functions, and three examples of how to prove the existence of (cyclic) subgroups for infinite families of sandpile groups by constructing appropriate integer-valued harmonic functions. The main open question concerns if the scaling-limits of the sandpile group for different sequences of polyforms converging to $\mathbb{R}^2$ are isomorphic.

math-ph↗
arXiv2018

Quantum groups as hidden symmetries of quantum impurities

We present an approach to interacting quantum many-body systems based on the notion of quantum groups, also known as $q$-deformed Lie algebras. In particular, we show that if the symmetry of a free quantum particle corresponds to a Lie group $G$, in the presence of a many-body environment this particle can be described by a deformed group, $G_q$. Crucially, the single deformation parameter, $q$, contains all the information about the many-particle interactions in the system. We exemplify our approach by considering a quantum rotor interacting with a bath of bosons, and demonstrate that extracting the value of $q$ from closed-form solutions in the perturbative regime allows one to predict the behavior of the system for arbitrary values of the impurity-bath coupling strength, in good agreement with non-perturbative calculations. Furthermore, the value of the deformation parameter allows to predict at which coupling strengths rotor-bath interactions result in a formation of a stable quasiparticle. The approach based on quantum groups does not only allow for a drastic simplification of impurity problems, but also provides valuable insights into hidden symmetries of interacting many-particle systems.

cond-mat.quant-gas↗
arXiv2018

Harmonic dynamics of the Abelian sandpile

The abelian sandpile serves as a model to study self-organized criticality, a phenomenon occurring in biological, physical and social processes. The identity of the abelian group is a fractal composed of self-similar patches, and its limit is subject of extensive collaborative research. Here, we analyze the evolution of the sandpile identity under harmonic fields of different orders. We show that this evolution corresponds to periodic cycles through the abelian group characterized by the smooth transformation and apparent conservation of the patches constituting the identity. The dynamics induced by second and third order harmonics resemble smooth stretchings, respectively translations, of the identity, while the ones induced by fourth order harmonics resemble magnifications and rotations. Starting with order three, the dynamics pass through extended regions of seemingly random configurations which spontaneously reassemble into accentuated patterns. We show that the space of harmonic functions projects to the extended analogue of the sandpile group, thus providing a set of universal coordinates identifying configurations between different domains. Since the original sandpile group is a subgroup of the extended one, this directly implies that it admits a natural renormalization. Furthermore, we show that the harmonic fields can be induced by simple Markov processes, and that the corresponding stochastic dynamics show remarkable robustness over hundreds of periods. Finally, we encode information into seemingly random configurations, and decode this information with an algorithm requiring minimal prior knowledge. Our results suggest that harmonic fields might split the sandpile group into sub-sets showing different critical coefficients, and that it might be possible to extend the fractal structure of the identity beyond the boundaries of its domain.

math-ph↗
arXiv2018

Self-Organized Criticality and Pattern Emergence through the lens of Tropical Geometry

Tropical Geometry, an established field in pure mathematics, is a place where String Theory, Mirror Symmetry, Computational Algebra, Auction Theory, etc, meet and influence each other. 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.

nlin.AO↗
arXiv2017

Sandpile solitons via smoothing of superharmonic functions

Let $F:\mathbb Z^2\to \mathbb Z$ 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 the 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.

math.CO↗
arXiv2017

Introduction to tropical series and wave dynamic on them

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.

math.CO↗
arXiv2015

Tropical curves in sandpiles

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 $μ\equiv 3$ is obtained by adding extra grains at several points. It appears, that the result $ψ^\circ$ of the relaxation of $ψ$ coincides with $μ$ almost everywhere; the set where $ψ^\circ\ne μ$ 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 $μ\equiv 3$. Le résultat $ψ^\circ$ de la relaxation est presque partout égal à $μ$. On appelle lieu de déviation l'ensemble des points où $ψ^\circ\ne μ$. 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.

math.CO↗
arXiv2015

Sandpiles on the heptagonal tiling

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 of such states is given.

math.CO↗