---
_id: '8325'
abstract:
- lang: eng
  text: "Let \U0001D439: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. "
acknowledgement: We thank Andrea Sportiello for sharing his insights on perturbative
  regimes of the Abelian sandpile model which was the starting point of our work.
  We also thank Grigory Mikhalkin, who encouraged us to approach this problem. We
  thank an anonymous referee. Also we thank Misha Khristoforov and Sergey Lanzat who
  participated on the initial state of this project, when we had nothing except the
  computer simulation and pictures. We thank Mikhail Raskin for providing us the code
  on Golly for faster simulations. Ilia Zharkov, Ilia Itenberg, Kristin Shaw, Max
  Karev, Lionel Levine, Ernesto Lupercio, Pavol Ševera, Yulieth Prieto, Michael Polyak,
  Danila Cherkashin asked us a lot of questions and listened to us; not all of their
  questions found answers here, but we are going to treat them in subsequent papers.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Nikita
  full_name: Kalinin, Nikita
  last_name: Kalinin
- first_name: Mikhail
  full_name: Shkolnikov, Mikhail
  id: 35084A62-F248-11E8-B48F-1D18A9856A87
  last_name: Shkolnikov
  orcid: 0000-0002-4310-178X
citation:
  ama: Kalinin N, Shkolnikov M. Sandpile solitons via smoothing of superharmonic functions.
    <i>Communications in Mathematical Physics</i>. 2020;378(9):1649-1675. doi:<a href="https://doi.org/10.1007/s00220-020-03828-8">10.1007/s00220-020-03828-8</a>
  apa: Kalinin, N., &#38; Shkolnikov, M. (2020). Sandpile solitons via smoothing of
    superharmonic functions. <i>Communications in Mathematical Physics</i>. Springer
    Nature. <a href="https://doi.org/10.1007/s00220-020-03828-8">https://doi.org/10.1007/s00220-020-03828-8</a>
  chicago: Kalinin, Nikita, and Mikhail Shkolnikov. “Sandpile Solitons via Smoothing
    of Superharmonic Functions.” <i>Communications in Mathematical Physics</i>. Springer
    Nature, 2020. <a href="https://doi.org/10.1007/s00220-020-03828-8">https://doi.org/10.1007/s00220-020-03828-8</a>.
  ieee: N. Kalinin and M. Shkolnikov, “Sandpile solitons via smoothing of superharmonic
    functions,” <i>Communications in Mathematical Physics</i>, vol. 378, no. 9. Springer
    Nature, pp. 1649–1675, 2020.
  ista: Kalinin N, Shkolnikov M. 2020. Sandpile solitons via smoothing of superharmonic
    functions. Communications in Mathematical Physics. 378(9), 1649–1675.
  mla: Kalinin, Nikita, and Mikhail Shkolnikov. “Sandpile Solitons via Smoothing of
    Superharmonic Functions.” <i>Communications in Mathematical Physics</i>, vol.
    378, no. 9, Springer Nature, 2020, pp. 1649–75, doi:<a href="https://doi.org/10.1007/s00220-020-03828-8">10.1007/s00220-020-03828-8</a>.
  short: N. Kalinin, M. Shkolnikov, Communications in Mathematical Physics 378 (2020)
    1649–1675.
date_created: 2020-08-30T22:01:13Z
date_published: 2020-09-01T00:00:00Z
date_updated: 2023-08-22T09:00:03Z
day: '01'
department:
- _id: TaHa
doi: 10.1007/s00220-020-03828-8
ec_funded: 1
external_id:
  arxiv:
  - '1711.04285'
  isi:
  - '000560620600001'
intvolume: '       378'
isi: 1
issue: '9'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1711.04285
month: '09'
oa: 1
oa_version: Preprint
page: 1649-1675
project:
- _id: 25681D80-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '291734'
  name: International IST Postdoc Fellowship Programme
publication: Communications in Mathematical Physics
publication_identifier:
  eissn:
  - '14320916'
  issn:
  - '00103616'
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Sandpile solitons via smoothing of superharmonic functions
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 378
year: '2020'
...
---
_id: '196'
abstract:
- lang: eng
  text: '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. '
acknowledgement: "M.L. is grateful to the members of the C Guet and G Tkacik groups
  for valuable comments and support. M.S. is grateful to Nikita Kalinin for inspiring
  communications.\r\n"
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Moritz
  full_name: Lang, Moritz
  id: 29E0800A-F248-11E8-B48F-1D18A9856A87
  last_name: Lang
- first_name: Mikhail
  full_name: Shkolnikov, Mikhail
  id: 35084A62-F248-11E8-B48F-1D18A9856A87
  last_name: Shkolnikov
  orcid: 0000-0002-4310-178X
citation:
  ama: Lang M, Shkolnikov M. Harmonic dynamics of the Abelian sandpile. <i>Proceedings
    of the National Academy of Sciences</i>. 2019;116(8):2821-2830. doi:<a href="https://doi.org/10.1073/pnas.1812015116">10.1073/pnas.1812015116</a>
  apa: Lang, M., &#38; Shkolnikov, M. (2019). Harmonic dynamics of the Abelian sandpile.
    <i>Proceedings of the National Academy of Sciences</i>. National Academy of Sciences.
    <a href="https://doi.org/10.1073/pnas.1812015116">https://doi.org/10.1073/pnas.1812015116</a>
  chicago: Lang, Moritz, and Mikhail Shkolnikov. “Harmonic Dynamics of the Abelian
    Sandpile.” <i>Proceedings of the National Academy of Sciences</i>. National Academy
    of Sciences, 2019. <a href="https://doi.org/10.1073/pnas.1812015116">https://doi.org/10.1073/pnas.1812015116</a>.
  ieee: M. Lang and M. Shkolnikov, “Harmonic dynamics of the Abelian sandpile,” <i>Proceedings
    of the National Academy of Sciences</i>, vol. 116, no. 8. National Academy of
    Sciences, pp. 2821–2830, 2019.
  ista: Lang M, Shkolnikov M. 2019. Harmonic dynamics of the Abelian sandpile. Proceedings
    of the National Academy of Sciences. 116(8), 2821–2830.
  mla: Lang, Moritz, and Mikhail Shkolnikov. “Harmonic Dynamics of the Abelian Sandpile.”
    <i>Proceedings of the National Academy of Sciences</i>, vol. 116, no. 8, National
    Academy of Sciences, 2019, pp. 2821–30, doi:<a href="https://doi.org/10.1073/pnas.1812015116">10.1073/pnas.1812015116</a>.
  short: M. Lang, M. Shkolnikov, Proceedings of the National Academy of Sciences 116
    (2019) 2821–2830.
date_created: 2018-12-11T11:45:08Z
date_published: 2019-02-19T00:00:00Z
date_updated: 2023-09-11T14:09:34Z
day: '19'
department:
- _id: CaGu
- _id: GaTk
- _id: TaHa
doi: 10.1073/pnas.1812015116
external_id:
  arxiv:
  - '1806.10823'
  isi:
  - '000459074400013'
  pmid:
  - ' 30728300'
intvolume: '       116'
isi: 1
issue: '8'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.1073/pnas.1812015116
month: '02'
oa: 1
oa_version: Published Version
page: 2821-2830
pmid: 1
publication: Proceedings of the National Academy of Sciences
publication_identifier:
  eissn:
  - 1091-6490
publication_status: published
publisher: National Academy of Sciences
quality_controlled: '1'
related_material:
  link:
  - description: News on IST Webpage
    relation: press_release
    url: https://ist.ac.at/en/news/famous-sandpile-model-shown-to-move-like-a-traveling-sand-dune/
scopus_import: '1'
status: public
title: Harmonic dynamics of the Abelian sandpile
type: journal_article
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 116
year: '2019'
...
---
_id: '441'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Nikita
  full_name: Kalinin, Nikita
  last_name: Kalinin
- first_name: Mikhail
  full_name: Shkolnikov, Mikhail
  id: 35084A62-F248-11E8-B48F-1D18A9856A87
  last_name: Shkolnikov
  orcid: 0000-0002-4310-178X
citation:
  ama: Kalinin N, Shkolnikov M. Tropical formulae for summation over a part of SL(2,Z).
    <i>European Journal of Mathematics</i>. 2019;5(3):909–928. doi:<a href="https://doi.org/10.1007/s40879-018-0218-0">10.1007/s40879-018-0218-0</a>
  apa: Kalinin, N., &#38; Shkolnikov, M. (2019). Tropical formulae for summation over
    a part of SL(2,Z). <i>European Journal of Mathematics</i>. Springer Nature. <a
    href="https://doi.org/10.1007/s40879-018-0218-0">https://doi.org/10.1007/s40879-018-0218-0</a>
  chicago: Kalinin, Nikita, and Mikhail Shkolnikov. “Tropical Formulae for Summation
    over a Part of SL(2,Z).” <i>European Journal of Mathematics</i>. Springer Nature,
    2019. <a href="https://doi.org/10.1007/s40879-018-0218-0">https://doi.org/10.1007/s40879-018-0218-0</a>.
  ieee: N. Kalinin and M. Shkolnikov, “Tropical formulae for summation over a part
    of SL(2,Z),” <i>European Journal of Mathematics</i>, vol. 5, no. 3. Springer Nature,
    pp. 909–928, 2019.
  ista: Kalinin N, Shkolnikov M. 2019. Tropical formulae for summation over a part
    of SL(2,Z). European Journal of Mathematics. 5(3), 909–928.
  mla: Kalinin, Nikita, and Mikhail Shkolnikov. “Tropical Formulae for Summation over
    a Part of SL(2,Z).” <i>European Journal of Mathematics</i>, vol. 5, no. 3, Springer
    Nature, 2019, pp. 909–928, doi:<a href="https://doi.org/10.1007/s40879-018-0218-0">10.1007/s40879-018-0218-0</a>.
  short: N. Kalinin, M. Shkolnikov, European Journal of Mathematics 5 (2019) 909–928.
date_created: 2018-12-11T11:46:29Z
date_published: 2019-09-15T00:00:00Z
date_updated: 2021-01-12T07:56:46Z
day: '15'
department:
- _id: TaHa
doi: 10.1007/s40879-018-0218-0
ec_funded: 1
external_id:
  arxiv:
  - '1711.02089'
intvolume: '         5'
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1711.02089
month: '09'
oa: 1
oa_version: Preprint
page: 909–928
project:
- _id: 25681D80-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '291734'
  name: International IST Postdoc Fellowship Programme
publication: European Journal of Mathematics
publication_identifier:
  eissn:
  - 2199-6768
  issn:
  - 2199-675X
publication_status: published
publisher: Springer Nature
publist_id: '7382'
quality_controlled: '1'
scopus_import: 1
status: public
title: Tropical formulae for summation over a part of SL(2,Z)
type: journal_article
user_id: D865714E-FA4E-11E9-B85B-F5C5E5697425
volume: 5
year: '2019'
...
---
_id: '303'
abstract:
- lang: eng
  text: 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.
acknowledgement: The first author, Nikita Kalinin, is funded by SNCF PostDoc.Mobility
  grant 168647. Support from the Basic Research Program of the National Research University
  Higher School of Economics is gratefully acknowledged. The second author, Mikhail
  Shkolnikov, is supported in part by the grant 159240 of the Swiss National Science
  Foundation as well as by the National Center of Competence in Research SwissMAP
  of the Swiss National Science Foundation.
article_processing_charge: No
arxiv: 1
author:
- first_name: Nikita
  full_name: Kalinin, Nikita
  last_name: Kalinin
- first_name: Mikhail
  full_name: Shkolnikov, Mikhail
  id: 35084A62-F248-11E8-B48F-1D18A9856A87
  last_name: Shkolnikov
  orcid: 0000-0002-4310-178X
citation:
  ama: Kalinin N, Shkolnikov M. Introduction to tropical series and wave dynamic on
    them. <i>Discrete and Continuous Dynamical Systems- Series A</i>. 2018;38(6):2827-2849.
    doi:<a href="https://doi.org/10.3934/dcds.2018120">10.3934/dcds.2018120</a>
  apa: Kalinin, N., &#38; Shkolnikov, M. (2018). Introduction to tropical series and
    wave dynamic on them. <i>Discrete and Continuous Dynamical Systems- Series A</i>.
    AIMS. <a href="https://doi.org/10.3934/dcds.2018120">https://doi.org/10.3934/dcds.2018120</a>
  chicago: Kalinin, Nikita, and Mikhail Shkolnikov. “Introduction to Tropical Series
    and Wave Dynamic on Them.” <i>Discrete and Continuous Dynamical Systems- Series
    A</i>. AIMS, 2018. <a href="https://doi.org/10.3934/dcds.2018120">https://doi.org/10.3934/dcds.2018120</a>.
  ieee: N. Kalinin and M. Shkolnikov, “Introduction to tropical series and wave dynamic
    on them,” <i>Discrete and Continuous Dynamical Systems- Series A</i>, vol. 38,
    no. 6. AIMS, pp. 2827–2849, 2018.
  ista: Kalinin N, Shkolnikov M. 2018. Introduction to tropical series and wave dynamic
    on them. Discrete and Continuous Dynamical Systems- Series A. 38(6), 2827–2849.
  mla: Kalinin, Nikita, and Mikhail Shkolnikov. “Introduction to Tropical Series and
    Wave Dynamic on Them.” <i>Discrete and Continuous Dynamical Systems- Series A</i>,
    vol. 38, no. 6, AIMS, 2018, pp. 2827–49, doi:<a href="https://doi.org/10.3934/dcds.2018120">10.3934/dcds.2018120</a>.
  short: N. Kalinin, M. Shkolnikov, Discrete and Continuous Dynamical Systems- Series
    A 38 (2018) 2827–2849.
date_created: 2018-12-11T11:45:43Z
date_published: 2018-06-01T00:00:00Z
date_updated: 2023-09-12T07:45:37Z
day: '01'
department:
- _id: TaHa
doi: 10.3934/dcds.2018120
external_id:
  arxiv:
  - '1706.03062'
  isi:
  - '000438818400007'
intvolume: '        38'
isi: 1
issue: '6'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1706.03062
month: '06'
oa: 1
oa_version: Submitted Version
page: 2827 - 2849
publication: Discrete and Continuous Dynamical Systems- Series A
publication_status: published
publisher: AIMS
publist_id: '7576'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Introduction to tropical series and wave dynamic on them
type: journal_article
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 38
year: '2018'
...
---
_id: '5794'
abstract:
- lang: eng
  text: 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.
article_number: '255302'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Enderalp
  full_name: Yakaboylu, Enderalp
  id: 38CB71F6-F248-11E8-B48F-1D18A9856A87
  last_name: Yakaboylu
  orcid: 0000-0001-5973-0874
- first_name: Mikhail
  full_name: Shkolnikov, Mikhail
  id: 35084A62-F248-11E8-B48F-1D18A9856A87
  last_name: Shkolnikov
  orcid: 0000-0002-4310-178X
- first_name: Mikhail
  full_name: Lemeshko, Mikhail
  id: 37CB05FA-F248-11E8-B48F-1D18A9856A87
  last_name: Lemeshko
  orcid: 0000-0002-6990-7802
citation:
  ama: Yakaboylu E, Shkolnikov M, Lemeshko M. Quantum groups as hidden symmetries
    of quantum impurities. <i>Physical Review Letters</i>. 2018;121(25). doi:<a href="https://doi.org/10.1103/PhysRevLett.121.255302">10.1103/PhysRevLett.121.255302</a>
  apa: Yakaboylu, E., Shkolnikov, M., &#38; Lemeshko, M. (2018). Quantum groups as
    hidden symmetries of quantum impurities. <i>Physical Review Letters</i>. American
    Physical Society. <a href="https://doi.org/10.1103/PhysRevLett.121.255302">https://doi.org/10.1103/PhysRevLett.121.255302</a>
  chicago: Yakaboylu, Enderalp, Mikhail Shkolnikov, and Mikhail Lemeshko. “Quantum
    Groups as Hidden Symmetries of Quantum Impurities.” <i>Physical Review Letters</i>.
    American Physical Society, 2018. <a href="https://doi.org/10.1103/PhysRevLett.121.255302">https://doi.org/10.1103/PhysRevLett.121.255302</a>.
  ieee: E. Yakaboylu, M. Shkolnikov, and M. Lemeshko, “Quantum groups as hidden symmetries
    of quantum impurities,” <i>Physical Review Letters</i>, vol. 121, no. 25. American
    Physical Society, 2018.
  ista: Yakaboylu E, Shkolnikov M, Lemeshko M. 2018. Quantum groups as hidden symmetries
    of quantum impurities. Physical Review Letters. 121(25), 255302.
  mla: Yakaboylu, Enderalp, et al. “Quantum Groups as Hidden Symmetries of Quantum
    Impurities.” <i>Physical Review Letters</i>, vol. 121, no. 25, 255302, American
    Physical Society, 2018, doi:<a href="https://doi.org/10.1103/PhysRevLett.121.255302">10.1103/PhysRevLett.121.255302</a>.
  short: E. Yakaboylu, M. Shkolnikov, M. Lemeshko, Physical Review Letters 121 (2018).
date_created: 2019-01-06T22:59:12Z
date_published: 2018-12-17T00:00:00Z
date_updated: 2023-09-15T12:09:06Z
day: '17'
department:
- _id: MiLe
doi: 10.1103/PhysRevLett.121.255302
ec_funded: 1
external_id:
  arxiv:
  - '1809.00222'
  isi:
  - '000454178600009'
intvolume: '       121'
isi: 1
issue: '25'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1809.00222
month: '12'
oa: 1
oa_version: Preprint
project:
- _id: 25681D80-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '291734'
  name: International IST Postdoc Fellowship Programme
- _id: 26031614-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: P29902
  name: Quantum rotations in the presence of a many-body environment
publication: Physical Review Letters
publication_identifier:
  issn:
  - '00319007'
publication_status: published
publisher: American Physical Society
quality_controlled: '1'
scopus_import: '1'
status: public
title: Quantum groups as hidden symmetries of quantum impurities
type: journal_article
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 121
year: '2018'
...
---
_id: '64'
abstract:
- lang: eng
  text: 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.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Nikita
  full_name: Kalinin, Nikita
  last_name: Kalinin
- first_name: Aldo
  full_name: Guzmán Sáenz, Aldo
  last_name: Guzmán Sáenz
- first_name: Y
  full_name: Prieto, Y
  last_name: Prieto
- first_name: Mikhail
  full_name: Shkolnikov, Mikhail
  id: 35084A62-F248-11E8-B48F-1D18A9856A87
  last_name: Shkolnikov
  orcid: 0000-0002-4310-178X
- first_name: V
  full_name: Kalinina, V
  last_name: Kalinina
- first_name: Ernesto
  full_name: Lupercio, Ernesto
  last_name: Lupercio
citation:
  ama: 'Kalinin N, Guzmán Sáenz A, Prieto Y, Shkolnikov M, Kalinina V, Lupercio E.
    Self-organized criticality and pattern emergence through the lens of tropical
    geometry. <i>PNAS: Proceedings of the National Academy of Sciences of the United
    States of America</i>. 2018;115(35):E8135-E8142. doi:<a href="https://doi.org/10.1073/pnas.1805847115">10.1073/pnas.1805847115</a>'
  apa: 'Kalinin, N., Guzmán Sáenz, A., Prieto, Y., Shkolnikov, M., Kalinina, V., &#38;
    Lupercio, E. (2018). Self-organized criticality and pattern emergence through
    the lens of tropical geometry. <i>PNAS: Proceedings of the National Academy of
    Sciences of the United States of America</i>. National Academy of Sciences. <a
    href="https://doi.org/10.1073/pnas.1805847115">https://doi.org/10.1073/pnas.1805847115</a>'
  chicago: 'Kalinin, Nikita, Aldo Guzmán Sáenz, Y Prieto, Mikhail Shkolnikov, V Kalinina,
    and Ernesto Lupercio. “Self-Organized Criticality and Pattern Emergence through
    the Lens of Tropical Geometry.” <i>PNAS: Proceedings of the National Academy of
    Sciences of the United States of America</i>. National Academy of Sciences, 2018.
    <a href="https://doi.org/10.1073/pnas.1805847115">https://doi.org/10.1073/pnas.1805847115</a>.'
  ieee: 'N. Kalinin, A. Guzmán Sáenz, Y. Prieto, M. Shkolnikov, V. Kalinina, and E.
    Lupercio, “Self-organized criticality and pattern emergence through the lens of
    tropical geometry,” <i>PNAS: Proceedings of the National Academy of Sciences of
    the United States of America</i>, vol. 115, no. 35. National Academy of Sciences,
    pp. E8135–E8142, 2018.'
  ista: 'Kalinin N, Guzmán Sáenz A, Prieto Y, Shkolnikov M, Kalinina V, Lupercio E.
    2018. Self-organized criticality and pattern emergence through the lens of tropical
    geometry. PNAS: Proceedings of the National Academy of Sciences of the United
    States of America. 115(35), E8135–E8142.'
  mla: 'Kalinin, Nikita, et al. “Self-Organized Criticality and Pattern Emergence
    through the Lens of Tropical Geometry.” <i>PNAS: Proceedings of the National Academy
    of Sciences of the United States of America</i>, vol. 115, no. 35, National Academy
    of Sciences, 2018, pp. E8135–42, doi:<a href="https://doi.org/10.1073/pnas.1805847115">10.1073/pnas.1805847115</a>.'
  short: 'N. Kalinin, A. Guzmán Sáenz, Y. Prieto, M. Shkolnikov, V. Kalinina, E. Lupercio,
    PNAS: Proceedings of the National Academy of Sciences of the United States of
    America 115 (2018) E8135–E8142.'
date_created: 2018-12-11T11:44:26Z
date_published: 2018-08-28T00:00:00Z
date_updated: 2023-09-18T08:41:16Z
day: '28'
department:
- _id: TaHa
doi: 10.1073/pnas.1805847115
ec_funded: 1
external_id:
  arxiv:
  - '1806.09153'
  isi:
  - '000442861600009'
intvolume: '       115'
isi: 1
issue: '35'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1806.09153
month: '08'
oa: 1
oa_version: Preprint
page: E8135 - E8142
project:
- _id: 25681D80-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '291734'
  name: International IST Postdoc Fellowship Programme
publication: 'PNAS: Proceedings of the National Academy of Sciences of the United
  States of America'
publication_identifier:
  issn:
  - '00278424'
publication_status: published
publisher: National Academy of Sciences
publist_id: '7990'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Self-organized criticality and pattern emergence through the lens of tropical
  geometry
type: journal_article
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 115
year: '2018'
...