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Lectures on the Ising and Potts Models on the Hypercubic Lattice

  • Hugo Duminil-CopinEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 304)

Abstract

Phase transitions are a central theme of statistical mechanics, and of probability more generally. Lattice spin models represent a general paradigm for phase transitions in finite dimensions, describing ferromagnets and even some fluids (lattice gases). It has been understood since the 1980s that random geometric representations, such as the random walk and random current representations, are powerful tools to understand spin models. In addition to techniques intrinsic to spin models, such representations provide access to rich ideas from percolation theory. In recent years, for two-dimensional spin models, these ideas have been further combined with ideas from discrete complex analysis. Spectacular results obtained through these connections include the proofs that interfaces of the two-dimensional Ising model have conformally invariant scaling limits given by SLE curves and the fact that the connective constant of the self-avoiding walk on the hexagonal lattice is given by \(\sqrt{2+\sqrt{2}}\). In higher dimensions, the understanding also progresses with the proof that the phase transition of Potts models is sharp, and that the magnetization of the three-dimensional Ising model vanishes at the critical point. These notes are largely inspired by [40, 42, 43].

Keywords

Statistical physics Phase transition Ising and potts model Percolation 

Notes

Acknowledgements

This research was funded by an IDEX Chair from Paris Saclay and by the NCCR SwissMap from the Swiss NSF. These lecture notes describe the content of a class given at the PIMS-CRM probability summer school on the behavior of lattice spin models near their critical point. The author would like to thank the organizers warmly for offering him the opportunity to give this course. Also, special thanks to people who sent comments to me, especially Timo Hirscher and Franco Severo.

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Institut des Hautes Études Scientifiques and Université de GenèveGenevaSwitzerland

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