@article{8325,
  abstract     = {Let 𝐹:ℤ2→ℤ be the pointwise minimum of several linear functions. The theory of smoothing allows us to prove that under certain conditions there exists the pointwise minimal function among all integer-valued superharmonic functions coinciding with F “at infinity”. We develop such a theory to prove existence of so-called solitons (or strings) in a 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 planar domains where solitons appear according to experiments. We prove that sandpile states, defined using our smoothing procedure, move changeless when we apply the wave operator (that is why we call them solitons), and can interact, forming triads and nodes. },
  author       = {Kalinin, Nikita and Shkolnikov, Mikhail},
  issn         = {14320916},
  journal      = {Communications in Mathematical Physics},
  number       = {9},
  pages        = {1649--1675},
  publisher    = {Springer Nature},
  title        = {{Sandpile solitons via smoothing of superharmonic functions}},
  doi          = {10.1007/s00220-020-03828-8},
  volume       = {378},
  year         = {2020},
}

@article{196,
  abstract     = {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. },
  author       = {Lang, Moritz and Shkolnikov, Mikhail},
  issn         = {1091-6490},
  journal      = {Proceedings of the National Academy of Sciences},
  number       = {8},
  pages        = {2821--2830},
  publisher    = {National Academy of Sciences},
  title        = {{Harmonic dynamics of the Abelian sandpile}},
  doi          = {10.1073/pnas.1812015116},
  volume       = {116},
  year         = {2019},
}

@article{441,
  author       = {Kalinin, Nikita and Shkolnikov, Mikhail},
  issn         = {2199-6768},
  journal      = {European Journal of Mathematics},
  number       = {3},
  pages        = {909–928},
  publisher    = {Springer Nature},
  title        = {{Tropical formulae for summation over a part of SL(2,Z)}},
  doi          = {10.1007/s40879-018-0218-0},
  volume       = {5},
  year         = {2019},
}

@article{303,
  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.},
  author       = {Kalinin, Nikita and Shkolnikov, Mikhail},
  journal      = {Discrete and Continuous Dynamical Systems- Series A},
  number       = {6},
  pages        = {2827 -- 2849},
  publisher    = {AIMS},
  title        = {{Introduction to tropical series and wave dynamic on them}},
  doi          = {10.3934/dcds.2018120},
  volume       = {38},
  year         = {2018},
}

@article{5794,
  abstract     = {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, Gq. 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 nonperturbative calculations. Furthermore, the value of the deformation parameter allows one 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.},
  author       = {Yakaboylu, Enderalp and Shkolnikov, Mikhail and Lemeshko, Mikhail},
  issn         = {00319007},
  journal      = {Physical Review Letters},
  number       = {25},
  publisher    = {American Physical Society},
  title        = {{Quantum groups as hidden symmetries of quantum impurities}},
  doi          = {10.1103/PhysRevLett.121.255302},
  volume       = {121},
  year         = {2018},
}

@article{64,
  abstract     = {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.},
  author       = {Kalinin, Nikita and Guzmán Sáenz, Aldo and Prieto, Y and Shkolnikov, Mikhail and Kalinina, V and Lupercio, Ernesto},
  issn         = {00278424},
  journal      = {PNAS: Proceedings of the National Academy of Sciences of the United States of America},
  number       = {35},
  pages        = {E8135 -- E8142},
  publisher    = {National Academy of Sciences},
  title        = {{Self-organized criticality and pattern emergence through the lens of tropical geometry}},
  doi          = {10.1073/pnas.1805847115},
  volume       = {115},
  year         = {2018},
}