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Lectures on the Ising and Potts Models on the Hypercubic Lattice

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Random Graphs, Phase Transitions, and the Gaussian Free Field (SSPROB 2017)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 304))

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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].

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Notes

  1. 1.

    One may also have \(\beta _c^\mathrm{exp}<\beta _c<\infty \), as shown in [70] for the planar clock model with \(q\gg 1\) states, but this situation is less common.

  2. 2.

    Kosterlitz and Thouless were awarded a Nobel prize in 2016 for their work on topological phase transitions.

  3. 3.

    The probability that \(\omega ^t_e=1\) is exactly the probability that \(\omega _e=1\) knowing the state of all the other edges.

  4. 4.

    Note that one may wish to pick \(K=3\) in (1.12) instead of a (a priori) larger K, but that this choice would make the construction of the trifurcations described below more difficult due to the fact that the three clusters may arrive very close to each other on the corner of \(\varLambda _n\), and therefore prevent us from “rewiring them” to construct a trifurcation at the origin.

  5. 5.

    The event \(\mathcal {A}_n{\setminus }\mathcal {H}_n\) is included in the event that there are two distinct clusters in \(R_n\) going from \(\varLambda _k\) to \(\partial R_n\). The intersection of the latter events for \(n\ge 1\) is included in the event that there are two distinct infinite clusters, which has zero probability. Thus, the probability of \(\mathcal {A}_n{\setminus } \mathcal {H}_n\) goes to 0 as n tends to infinity.

  6. 6.

    A sequence \((f_n)\) of continuous homeomorphisms from [0, 1] onto itself satisfies a sharp threshold if for any \(\varepsilon >0\), \(\varDelta _n(\varepsilon ):=f_n^{-1}(1-\varepsilon )-f_n^{-1}(\varepsilon )\) tends to 0.

  7. 7.

    Formally, we only obtained the result for n even, but the result for n odd can be obtained similarly.

  8. 8.

    Recall that \((\omega ^*)^1\) is the graph \(\omega ^*\) where all vertices of \(\partial G^*\) are identified together. This graph can clearly be embedded in the plane by “moving” the vertices of \(\partial G^*\) to a single point chosen in the exterior face of \(\omega \), and drawing the edges incident to \(\partial G^*\) by “extending” the corresponding edges of \(\omega ^*\) by continuous curves not intersecting each other or edges of \(\omega \), and going to this chosen point.

  9. 9.

    Truncated correlations is a vague term referring to differences of correlation functions (for instance \(\mu ^+_\beta [\sigma _x\sigma _y]-\mu ^+_\beta [\sigma _x]\mu ^+_\beta [\sigma _y]\) or \(\mu ^+_\beta [\sigma _x\sigma _y]-\mu ^\mathrm{f}_\beta [\sigma _x\sigma _y]\) or \(U_4(x_1,x_2,x_3,x_4)\) defined later in this section).

  10. 10.

    When \(A=\{x,y\}\), the event \(\mathcal {F}_A\) is simply the event that x and y are connected to each other.

  11. 11.

    Meaning that these paths start and end in B, and each element in B appears exactly once in the set of beginning and ends of these paths.

  12. 12.

    Or more elegantly the use of Exercise 5, which states that \(\phi ^0_{p,2}[\omega _{xy}]=p\cdot \mu ^\mathrm{f}_{\beta }[\sigma _x=\sigma _y]\), which combined with \(\mu ^\mathrm{f}_{\beta }[\sigma _x=\sigma _y]=2\mu ^\mathrm{f}_{\beta }[\sigma _x\sigma _y]-1=2\phi ^0_{p,2}[x\leftrightarrow y]-1\) gives the requested equality.

  13. 13.

    Note that they are not equal to the mix boundary conditions since \(\gamma \) and \(\gamma '\) are wired together.

  14. 14.

    i.e. a path of edges starting and ending at the same point.

  15. 15.

    Let us mention that there are other instances of parafermionic observables for the self-avoiding walk, see [9, 72]. We do not discuss this further here since our goal is to quickly move back to the random-cluster model.

  16. 16.

    The exploration path \(\gamma \) is considered as a loop and counts as 1 in \(\ell (\overline{\omega })\).

  17. 17.

    Fermions have half-integer spins while bosons have integer spins, there are no particles with fractional spin, but the use of such fractional spins at a theoretical level has been very fruitful in physics.

  18. 18.

    We did not prove that P4b implies P5, but since P4a implies P4b and P5, this follows readily.

  19. 19.

    If it has “pinched” points, we add a tiny ball of size \(\varepsilon \ll \delta \). The very precise definition is not relevant here since the definition is a complicated way of phrasing an intuitive notion of convergence.

  20. 20.

    Let us mention crossing probabilities [15, 91], interfaces with different boundary conditions [35, 82], full family of interfaces [16, 94], the energy fields [81, 83]. The observable has also been used off criticality, see [12, 48].

  21. 21.

    Recall that here and below, we consider the convergence on every compact subset of \(\varvec{\Omega }\).

  22. 22.

    The proof of the existence of this map is not completely obvious and requires Schwarz’s reflection principle.

  23. 23.

    Again, one usually requires a few things about this function, but let us omit these technical conditions here.

  24. 24.

    Since Wick’s rule is equivalent to the fact that \(U_4(x_1,x_2,x_3,x_4)\) vanishes (see the definition in Exercise 35), this quantity is a measure of how non-Gaussian the field \((\sigma _x:x\in \mathbb Z^d)\) is.

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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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  1. Institut des Hautes Études Scientifiques and Université de Genève, Geneva, Switzerland

    Hugo Duminil-Copin

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Correspondence to Hugo Duminil-Copin .

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  1. Department of Mathematics, The University of British Columbia, Vancouver, BC, Canada

    Martin T. Barlow

  2. Department of Mathematics, The University of British Columbia, Vancouver, BC, Canada

    Gordon Slade

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Duminil-Copin, H. (2020). Lectures on the Ising and Potts Models on the Hypercubic Lattice. In: Barlow, M., Slade, G. (eds) Random Graphs, Phase Transitions, and the Gaussian Free Field. SSPROB 2017. Springer Proceedings in Mathematics & Statistics, vol 304. Springer, Cham. https://doi.org/10.1007/978-3-030-32011-9_2

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