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Emergent planarity in two-dimensional Ising models with finite-range Interactions

  • Michael AizenmanEmail author
  • Hugo Duminil-Copin
  • Vincent Tassion
  • Simone Warzel
Article
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Abstract

The known Pfaffian structure of the boundary spin correlations, and more generally order–disorder correlation functions, is given a new explanation through simple topological considerations within the model’s random current representation. This perspective is then employed in the proof that the Pfaffian structure of boundary correlations emerges asymptotically at criticality in Ising models on \({\mathbb {Z}}^2\) with finite-range interactions. The analysis is enabled by new results on the stochastic geometry of the corresponding random currents. The proven statement establishes an aspect of universality, seen here in the emergence of fermionic structures in two dimensions beyond the solvable cases.

Notes

Acknowledgements

This work was supported in parts by the NSF Grants PHY-1305472, DMS-1613296, the NCCR SwissMAP, the IDEX grant from Paris-Saclay, the ERC CriBLaM, and a Princeton University/University of Geneva partnership fund.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Michael Aizenman
    • 1
    Email author
  • Hugo Duminil-Copin
    • 2
    • 3
  • Vincent Tassion
    • 4
  • Simone Warzel
    • 5
  1. 1.Princeton UniversityPrincetonUSA
  2. 2.Institut des Hautes Études ScientifiquesBures-sur-YvetteFrance
  3. 3.Université de GenèveGenevaSwitzerland
  4. 4.ETH ZürichZurichSwitzerland
  5. 5.Technische Universität MünchenGarchingGermany

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