Skip to main content
SpringerLink
Log in

Lectures on the Spin and Loop O(n) Models

  • Conference paper
  • First Online:
Sojourns in Probability Theory and Statistical Physics - I

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 298))

Abstract

The classical spin O(n) model is a model on a d-dimensional lattice in which a vector on the \((n-1)\)-dimensional sphere is assigned to every lattice site and the vectors at adjacent sites interact ferromagnetically via their inner product. Special cases include the Ising model (\(n=1\)), the XY model (\(n=2\)) and the Heisenberg model (\(n=3\)). We discuss questions of long-range order and decay of correlations in the spin O(n) model for different combinations of the lattice dimension d and the number of spin components n.

The loop O(n) model is a model for a random configuration of disjoint loops. We discuss its properties on the hexagonal lattice. The model is parameterized by a loop weight \(n\ge 0\) and an edge weight \(x\ge 0\). Special cases include self-avoiding walk (\(n=0\)), the Ising model (\(n=1\)), critical percolation (\(n=x=1\)), dimer model (\(n=1,x=\infty \)), proper 4-coloring (\(n=2, x=\infty )\), integer-valued (\(n=2\)) and tree-valued (integer \(n>=3\)) Lipschitz functions and the hard hexagon model (\(n=\infty \)). The object of study in the model is the typical structure of loops. We review the connection of the model with the spin O(n) model and discuss its conjectured phase diagram, emphasizing the many open problems remaining.

R. Peled and Y. Spinka—Research supported by Israeli Science Foundation grant 861/15 and the European Research Council starting grant 678520 (LocalOrder).

Y. Spinka—Research supported by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Exponential decay is stated in these references in the infinite-volume limit, but is derived as a consequence of a finite-volume criterion and is thus implied, as the infinite-volume measure is unique, also in finite volume.

  2. 2.

    A related intuition was mentioned earlier by Herring and Kittel [68, Footnote 8a].

  3. 3.

    In fact, more is true, conditioned on \((\nabla \theta _{=k})_{1 \le k \le m}\), the \(\sigma \)-algebras of \(\nabla \theta _{\ell -1 \le \cdot \le \ell }\) are independent for \(1 \le \ell \le m\), where \(\nabla \theta _{\ell -1 \le \cdot \le \ell }\) is the collection of gradients \(\theta _u - \theta _v\) with \(2^{\ell -1} \le \Vert u\Vert _1,\Vert v\Vert _1 \le 2^\ell \).

  4. 4.

    It suffices to show that \(\iint \prod _{i,j=1}^n h(s_i,t_j)d\lambda (s_i)d\lambda (t_j)>0\) for \(n\ge 1\). Fubini’s theorem reduces this to \(\iint \prod _{i=1}^n h(s_i,t)d\lambda (s_i)d\lambda (t)>0\), which then follows from Fubini’s theorem and the assumption on h.

References

  1. Aizenman, M.: Absence of an intermediate phase for a general class of one-component ferromagnetic models. Phys. Rev. Lett. 54(8), 839 (1985)

    Article  Google Scholar 

  2. Aizenman, M.: Rigorous studies of critical behavior. II. In: Statistical Physics and Dynamical Systems (Köszeg, 1984). Progress in Physics, vol. 10, pp. 453–481. Birkhäuser, Boston (1985)

    Google Scholar 

  3. Aizenman, M.: On the slow decay of \({\rm O}(2)\) correlations in the absence of topological excitations: remark on the Patrascioiu-Seiler model. J. Statist. Phys. 77(1–2), 351–359 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. Aizenman, M., Barsky, D.J., Fernández, R.: The phase transition in a general class of Ising-type models is sharp. J. Statist. Phys. 47(3–4), 343–374 (1987)

    Article  MathSciNet  Google Scholar 

  5. Aizenman, M., Bricmont, J., Lebowitz, J.: Percolation of the minority spins in high-dimensional Ising models. J. Stat. Phys. 49(3), 859–865 (1987)

    Article  MATH  Google Scholar 

  6. Aizenman, M., Duminil-Copin, H., Sidoravicius, V.: Random currents and continuity of Ising model’s spontaneous magnetization. Commun. Math. Phys. 334(2), 719–742 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Andrews, G.E., Baxter, R.J., Forrester, P.J.: Eight-vertex sos model and generalized Rogers-Ramanujan-type identities. J. Stat. Phys. 35(3), 193–266 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  8. Balaban, T.: A low temperature expansion for classical N-vector models. I. A renormalization group flow. Commun. Math. Phys. 167(1), 103–154 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  9. Balister, P.N., Bollobás, B.: Counting regions with bounded surface area. Commun. Math. Phys. 273(2), 305–315 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bauerschmidt, R.: Ferromagnetic spin systems (2016). Lecture notes: http://www.statslab.cam.ac.uk/~rb812/doc/spin.pdf

  11. Baxter, R.: Colorings of a hexagonal lattice. J. Math. Phys. 11(3), 784–789 (1970)

    Article  MathSciNet  Google Scholar 

  12. Baxter, R.J.: Hard hexagons: exact solution. J. Phys. A 13(3), L61–L70 (1980)

    Article  MathSciNet  Google Scholar 

  13. Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London (1989). Reprint of the 1982 original

    MATH  Google Scholar 

  14. Beffara, V., Gayet, D.: Percolation without FKG. Preprint arXiv:1710.10644 (2017)

  15. Benassi, C., Lees, B., Ueltschi, D.: Correlation inequalities for classical and quantum XY models. Preprint arXiv:1611.06019 (2016)

  16. Benoist, S., Hongler, C.: The scaling limit of critical Ising interfaces is CLE(3). arXiv preprint arXiv:1604.06975 (2016)

  17. Berezinskii, V.: Destruction of long-range order in one-dimensional and two-dimensional systems possessing a continuous symmetry group. II. Quantum systems. Sov. J. Exp. Theor. Phys. 34, 610 (1972)

    Google Scholar 

  18. Berlin, T.H., Kac, M.: The spherical model of a ferromagnet. Phys. Rev. 86(6), 821 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  19. Biskup, M.: Reflection positivity and phase transitions in lattice spin models. In: Methods of Contemporary Mathematical Statistical Physics, pp. 1–86 (2009)

    Google Scholar 

  20. Bollobás, B.: The Art of Mathematics. Cambridge University Press, New York (2006)

    Book  MATH  Google Scholar 

  21. Bonato, C., Perez, J.F., Klein, A.: The Mermin-Wagner phenomenon and cluster properties of one-and two-dimensional systems. J. Stat. Phys. 29(2), 159–175 (1982)

    Article  MathSciNet  Google Scholar 

  22. Bricmont, J., Fontaine, J., Landau, L.: On the uniqueness of the equilibrium state for plane rotators. Commun. Math. Phys. 56(3), 281–296 (1977)

    Article  MathSciNet  Google Scholar 

  23. Brydges, D., Fröhlich, J., Spencer, T.: The random walk representation of classical spin systems and correlation inequalities. Commun. Math. Phys. 83(1), 123–150 (1982)

    Article  MathSciNet  Google Scholar 

  24. Camia, F., Newman, C.: Critical percolation exploration path and \({\rm SLE}_6\): a proof of convergence. Probab. Theory Relat. Fields 139(3–4), 473–519 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Camia, F., Newman, C.M.: Continuum nonsimple loops and 2D critical percolation. J. Stat. Phys. 116(1), 157–173 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Camia, F., Newman, C.M.: The full scaling limit of two-dimensional critical percolation. Preprint arXiv:math/0504036 (2005)

  27. Camia, F., Newman, C.M.: Two-dimensional critical percolation: the full scaling limit. Commun. Math. Phys. 268(1), 1–38 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  28. Cardy, J.: Conformal field theory and statistical mechanics. In: Exact Methods in Low-Dimensional Statistical Physics and Quantum Computing, pp. 65–98 (2008)

    Google Scholar 

  29. Chayes, L., Pryadko, L.P., Shtengel, K.: Intersecting loop models on \(\mathbb{Z}^d\): rigorous results. Nucl. Phys. B 570(3), 590–614 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  30. Chelkak, D., Duminil-Copin, H., Hongler, C., Kemppainen, A., Smirnov, S.: Convergence of Ising interfaces to Schramm’s SLE curves. C. R. Math. Acad. Sci. Paris 352(2), 157–161 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Chelkak, D., Hongler, C., Izyurov, K.: Conformal invariance of spin correlations in the planar Ising model. Ann. Math. (2) 181(3), 1087–1138 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. Chelkak, D., Smirnov, S.: Universality in the 2D Ising model and conformal invariance of fermionic observables. Invent. Math. 189(3), 515–580 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. Cohen-Alloro, O., Peled, R.: Rarity of extremal edges in random surfaces and other theoretical applications of cluster algorithms. Preprint arXiv:1711.00259 (2017)

  34. Crawford, N., Glazman, A., Harel, M., Peled, R.: Macroscopic loops in the loop \(O(n)\) model via the XOR trick (in preparation)

    Google Scholar 

  35. Dimock, J.: The renormalization group according to Balaban I: Small fields. Rev. Math. Phys. 25(07), 1330,010 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  36. Dobrushin, R., Shlosman, S.: Absence of breakdown of continuous symmetry in two-dimensional models of statistical physics. Commun. Math. Phys. 42(1), 31–40 (1975)

    Article  MathSciNet  Google Scholar 

  37. Domany, E., Mukamel, D., Nienhuis, B., Schwimmer, A.: Duality relations and equivalences for models with O(n) and cubic symmetry. Nuclear Phys. B 190(2), 279–287 (1981)

    Article  Google Scholar 

  38. Duminil-Copin, H.: Parafermionic observables and their applications to planar statistical physics models. Ensaios Matematicos 25, 1–371 (2013)

    MathSciNet  MATH  Google Scholar 

  39. Duminil-Copin, H., Glazman, A., Peled, R., Spinka, Y.: Macroscopic loops in the loop \(O(n)\) model at Nienhuis’ critical point. Preprint arXiv:1707.09335 (2017). J. Eur. Math. Soc. (2017, to appear)

  40. Duminil-Copin, H., Kozma, G., Yadin, A.: Supercritical self-avoiding walks are space-filling. Ann. Inst. H. Poincaré Probab. Stat. 50(2), 315–326 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  41. Duminil-Copin, H., Peled, R., Samotij, W., Spinka, Y.: Exponential decay of loop lengths in the loop \(O(n)\) model with large \(n\). Commun. Math. Phys. 349, 777–817 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  42. Duminil-Copin, H., Raoufi, A., Tassion, V.: Sharp phase transition for the random-cluster and Potts models via decision trees. Preprint arXiv:1705.03104 (2017). Ann. Math. (2017, published)

  43. Duminil-Copin, H., Sidoravicius, V., Tassion, V.: Continuity of the phase transition for planar Potts models with \(1\le q\le 4\). Commun. Math. Phys. 349(1), 47–107 (2017)

    Article  MATH  Google Scholar 

  44. Duminil-Copin, H., Smirnov, S.: The connective constant of the honeycomb lattice equals \(\sqrt{2+\sqrt{2}}\). Ann. Math. (2) 175(3), 1653–1665 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  45. Duminil-Copin, H., Tassion, V.: A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. Commun. Math. Phys. 343(2), 725–745 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  46. Dyson, F.J., Lieb, E.H., Simon, B.: Phase transitions in quantum spin systems with isotropic and nonisotropic interactions. In: Nachtergaele, B., Solovej, J.P., Yngvason, J. (eds.) Statistical Mechanics, pp. 163–211. Springer, Heidelberg (1978)

    Chapter  Google Scholar 

  47. Edwards, R.G., Sokal, A.D.: Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm. Phys. Rev. D (3) 38(6), 2009–2012 (1988)

    Article  MathSciNet  Google Scholar 

  48. Fernández, R., Fröhlich, J., Sokal, A.D.: Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory. Springer, Heidelberg (2013)

    MATH  Google Scholar 

  49. Fisher, M.E.: Critical temperatures of anisotropic Ising lattices. II. General upper bounds. Phys. Rev. 162(2), 480 (1967)

    Article  Google Scholar 

  50. Friedli, S., Velenik, Y.: Statistical mechanics of lattice systems: a concrete mathematical introduction. Cambridge University Press (2017, to appear). http://www.unige.ch/math/folks/velenik/smbook

  51. Fröhlich, J., Israel, R., Lieb, E.H., Simon, B.: Phase transitions and reflection positivity. I. General theory and long range lattice models. In: Nachtergaele, B., Solovej, J.P., Yngvason, J. (eds.) Statistical Mechanics, pp. 213–246. Springer, Heidelberg (1978)

    Chapter  Google Scholar 

  52. Fröhlich, J., Israel, R.B., Lieb, E.H., Simon, B.: Phase transitions and reflection positivity. II. Lattice systems with short-range and Coulomb interactions. In: Nachtergaele, B., Solovej, J.P., Yngvason, J. (eds.) Statistical Mechanics, pp. 247–297. Springer, Heidelberg (1980)

    Chapter  Google Scholar 

  53. Fröhlich, J., Simon, B., Spencer, T.: Infrared bounds, phase transitions and continuous symmetry breaking. Commun. Math. Phys. 50(1), 79–95 (1976)

    Article  MathSciNet  Google Scholar 

  54. Fröhlich, J., Spencer, T.: The Kosterlitz-Thouless transition in two-dimensional Abelian spin systems and the Coulomb gas. Commun. Math. Phys. 81(4), 527–602 (1981). http://projecteuclid.org/getRecord?id=euclid.cmp/1103920388

    Article  MathSciNet  Google Scholar 

  55. Fröhlich, J., Spencer, T.: Massless phases and symmetry restoration in abelian gauge theories and spin systems. Commun. Math. Phys. 83(3), 411–454 (1982)

    Article  MathSciNet  Google Scholar 

  56. Gagnebin, M., Miłoś, P., Peled, R.: In preparation

    Google Scholar 

  57. Gagnebin, M., Velenik, Y.: Upper bound on the decay of correlations in a general class of O(N)-symmetric models. Commun. Math. Phys. 332(3), 1235–1255 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  58. Georgii, H.O., Higuchi, Y.: Percolation and number of phases in the two-dimensional Ising model. J. Math. Phys. 41(3), 1153–1169 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  59. Ginibre, J.: Simple proof and generalization of Griffiths’ second inequality. Phys. Rev. Lett. 23(15), 828 (1969)

    Article  Google Scholar 

  60. Ginibre, J.: General formulation of Griffiths’ inequalities. Commun. Math. Phys. 16, 310–328 (1970)

    Article  MathSciNet  Google Scholar 

  61. Glazman, A., Manolescu, I.: Exponential decay in the loop \(O(n)\) model: \(n\,>\, 1\), \(x\,<\,\tfrac{1}{\sqrt{3}}+\varepsilon (n)\). arXiv preprint arXiv:1810.11302 (2018)

  62. Glazman, A., Manolescu, I.: Uniform Lipschitz functions on the triangular lattice have logarithmic variations. arXiv preprint arXiv:1810.05592 (2018)

  63. Griffiths, R.: Correlation in Ising ferromagnets I, II. J. Math. Phys. 8, 478–489 (1967)

    Article  Google Scholar 

  64. Griffiths, R.B.: Correlations in Ising ferromagnets. III. Commun. Math. Phys. 6(2), 121–127 (1967)

    Article  Google Scholar 

  65. Grimmett, G.: The Random-Cluster Model. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 333. Springer, Berlin (2006)

    Book  MATH  Google Scholar 

  66. Häggström, O., Jonasson, J., et al.: Uniqueness and non-uniqueness in percolation theory. Probab. Surv. 3, 289–344 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  67. Hasenbusch, M.: The two-dimensional XY model at the transition temperature: a high-precision Monte Carlo study. J. Phys. A Math. Gen. 38(26), 5869 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  68. Herring, C., Kittel, C.: On the theory of spin waves in ferromagnetic media. Phys. Rev. 81(5), 869 (1951)

    Article  MATH  Google Scholar 

  69. Hohenberg, P.: Existence of long-range order in one and two dimensions. Phys. Rev. 158(2), 383 (1967)

    Article  Google Scholar 

  70. Hongler, C., Kytölä, K.: Ising interfaces and free boundary conditions. J. Amer. Math. Soc. 26(4), 1107–1189 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  71. Hurst, C., Sherman, S.: Griffiths’ theorems for the ferromagnetic Heisenberg model. Phys. Rev. Lett. 22(25), 1357 (1969)

    Article  Google Scholar 

  72. Ioffe, D., Shlosman, S., Velenik, Y.: 2D models of statistical physics with continuous symmetry: the case of singular interactions. Commun. Math. Phys. 226(2), 433–454 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  73. Ito, K.: Clustering in low-dimensional SO(N)-invariant statistical models with long-range interactions. J. Stat. Phys. 29(4), 747–760 (1982)

    Article  MathSciNet  Google Scholar 

  74. Izyurov, K.: Smirnov’s observable for free boundary conditions, interfaces and crossing probabilities. Commun. Math. Phys. 337(1), 225–252 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  75. Jasnow, D., Fisher, M.E.: Broken symmetry and decay of order in restricted dimensionality. Phys. Rev. Lett. 23(6), 286 (1969)

    Article  MathSciNet  Google Scholar 

  76. Kac, M., Thompson, C.J.: Spherical model and the infinite spin dimensionality limit. Phys. Norveg. 5(3–4), 163–168 (1971)

    MathSciNet  Google Scholar 

  77. Kager, W., Nienhuis, B.: A guide to stochastic Löwner evolution and its applications. J. Stat. Phys. 115(5–6), 1149–1229 (2004)

    Article  MATH  Google Scholar 

  78. Kennedy, T., King, C.: Spontaneous symmetry breakdown in the Abelian Higgs model. Commun. Math. Phys. 104(2), 327–347 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  79. Kharash, V., Peled, R.: The Fröhlich-Spencer proof of the Berezinskii-Kosterlitz-Thouless transition. Preprint arXiv:1711.04720 (2017)

  80. Komura, Y., Okabe, Y.: Large-scale Monte Carlo simulation of two-dimensional classical XY model using multiple GPUs. J. Phys. Soc. Jpn. 81(11), 113,001 (2012)

    Article  Google Scholar 

  81. Kosterlitz, J.M., Thouless, D.J.: Long range order and metastability in two dimensional solids and superfluids. (Application of dislocation theory). J. Phys. C Solid State Phys. 5, L124–L126 (1972)

    Article  Google Scholar 

  82. Kosterlitz, J.M., Thouless, D.J.: Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C Solid State Phys. 6(7), 1181–1203 (1973)

    Article  Google Scholar 

  83. Kramers, H.A., Wannier, G.H.: Statistics of the two-dimensional ferromagnet. I. Phys. Rev. 2(60), 252–262 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  84. Kupiainen, A.J.: On the \(1/n\) expansion. Commun. Math. Phys. 73(3), 273–294 (1980). http://projecteuclid.org/getRecord?id=euclid.cmp/1103907876

    Article  MathSciNet  Google Scholar 

  85. Lebowitz, J.L., Mazel, A.E.: Improved Peierls argument for high-dimensional Ising models. J. Stat. Phys. 90(3–4), 1051–1059 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  86. McBryan, O.A., Spencer, T.: On the decay of correlations in \({\rm SO}(n)\)-symmetric ferromagnets. Commun. Math. Phys. 53(3), 299–302 (1977)

    Article  MathSciNet  Google Scholar 

  87. McCoy, B., Wu, T.: The Two-Dimensional Ising Model. Harvard University Press, Cambridge (1973)

    Book  MATH  Google Scholar 

  88. Mermin, N.D.: Absence of ordering in certain classical systems. J. Math. Phys. 8(5), 1061–1064 (1967)

    Article  Google Scholar 

  89. Mermin, N.D., Wagner, H.: Absence of ferromagnetism or antiferromagnetism in one-or two-dimensional isotropic Heisenberg models. Phys. Rev. Lett. 17(22), 1133 (1966)

    Article  Google Scholar 

  90. Messager, A., Miracle-Sole, S., Ruiz, J.: Upper bounds on the decay of correlations in SO(N)-symmetric spin systems with long range interactions. Ann. Inst. H. Poincaré Sect. A (NS) 40(1), 85–96 (1984)

    MathSciNet  Google Scholar 

  91. Miłoś, P., Peled, R.: Delocalization of two-dimensional random surfaces with hard-core constraints. Commun. Math. Phys. 340(1), 1–46 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  92. Naddaf, A.: On the decay of correlations in non-analytic SO(n)-symmetric models. Commun. Math. Phys. 184(2), 387–395 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  93. Nienhuis, B.: Exact critical point and critical exponents of \(\rm O(n)\) models in two dimensions. Phys. Rev. Lett. 49(15), 1062–1065 (1982)

    Article  MathSciNet  Google Scholar 

  94. Nienhuis, B.: Locus of the tricritical transition in a two-dimensional q-state Potts model. Phys. A Stat. Mech. Its Appl. 177(1–3), 109–113 (1991)

    Article  MathSciNet  Google Scholar 

  95. Onsager, L.: Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. (2) 65, 117–149 (1944)

    Article  MathSciNet  MATH  Google Scholar 

  96. Patrascioiu, A., Seiler, E.: Phase structure of two-dimensional spin models and percolation. J. Stat. Phys. 69(3–4), 573–595 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  97. Peierls, R.: On Ising’s model of ferromagnetism. Math. Proc. Camb. Philos. Soc. 32, 477–481 (1936)

    Article  MATH  Google Scholar 

  98. Peled, R.: High-dimensional Lipschitz functions are typically flat. Ann. Probab. 45(3), 1351–1447 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  99. Pfister, C.E.: On the symmetry of the Gibbs states in two dimensional lattice systems. Commun. Math. Phys. 79(2), 181–188 (1981)

    Article  MathSciNet  Google Scholar 

  100. Pinson, H.: Rotational invariance of the 2D spin-spin correlation function. Commun. Math. Phys. 314(3), 807–816 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  101. Polyakov, A.M.: Interaction of Goldstone particles in two dimensions. Applications to ferromagnets and massive Yang-Mills fields. Phys. Lett. B 59(1), 79–81 (1975)

    Article  Google Scholar 

  102. Richthammer, T.: Translation-invariance of two-dimensional Gibbsian point processes. Commun. Math. Phys. 274(1), 81–122 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  103. Sakai, A.: Lace expansion for the Ising model. Commun. Math. Phys. 272(2), 283–344 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  104. Schramm, O.: Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118, 221–288 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  105. Sheffield, S.: Exploration trees and conformal loop ensembles. Duke Math. J. 147(1), 79–129 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  106. Shlosman, S.B.: Absence of continuous symmetry breaking in two-dimensional models of statistical physics. Theor. Math. Phys. 33(1), 897–902 (1977)

    Article  Google Scholar 

  107. Shlosman, S.B.: Decrease of correlations in two-dimensional models with continuous symmetry group. Theor. Math. Phys. 37(3), 1118–1120 (1978)

    Article  Google Scholar 

  108. Simon, B.: Mean field upper bound on the transition temperature in multicomponent ferromagnets. J. Stat. Phys. 22(4), 491–493 (1980)

    Article  MathSciNet  Google Scholar 

  109. Simon, B., Sokal, A.D.: Rigorous entropy-energy arguments. J. Stat. Phys. 25(4), 679–694 (1981)

    Article  MathSciNet  Google Scholar 

  110. Smirnov, S.: Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333(3), 239–244 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  111. Smirnov, S.: Towards conformal invariance of 2D lattice models. In: International Congress of Mathematicians, vol. II, pp. 1421–1451. European Mathematical Society, Zürich (2006)

    Google Scholar 

  112. Smirnov, S.: Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. Math. (2) 172(2), 1435–1467 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  113. Stanley, H.E.: Spherical model as the limit of infinite spin dimensionality. Phys. Rev. 176, 718–722 (1968)

    Article  Google Scholar 

  114. Sylvester, G.S.: The Ginibre inequality. Commun. Math. Phys. 73(2), 105–114 (1980)

    Article  MathSciNet  Google Scholar 

  115. Symanzik, K.: Euclidean quantum field theory. In: Proceedings of the 45th International School of Physics ‘Enrico Fermi’: Local Quantum Theory. Acaedmic Press, New York, London (1969)

    Google Scholar 

  116. Taggi, L.: Shifted critical threshold in the loop \(O(n)\) model at arbitrary small \(n\). Preprint arXiv:1806.09360 (2018)

  117. Tassion, V.: Crossing probabilities for Voronoi percolation. Ann. Probab. 44(5), 3385–3398 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  118. Timár, Á.: Boundary-connectivity via graph theory. Proc. Amer. Math. Soc. 141(2), 475–480 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  119. Ueltschi, D.: Quantum spin systems and phase transitions (2013). Lecture notes: http://www.ueltschi.org/publications.php

  120. Wolff, U.: Collective Monte Carlo updating for spin systems. Phys. Rev. Lett. 62(4), 361 (1989)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel

    Ron Peled & Yinon Spinka

Authors
  1. Ron Peled
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yinon Spinka
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ron Peled .

Editor information

Editors and Affiliations

  1. NYU Shanghai, Shanghai, China

    Vladas Sidoravicius

Additional information

Dedicated to Chuck Newman on the occasion of his 70th birthday

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Singapore Pte Ltd.

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Peled, R., Spinka, Y. (2019). Lectures on the Spin and Loop O(n) Models. In: Sidoravicius, V. (eds) Sojourns in Probability Theory and Statistical Physics - I. Springer Proceedings in Mathematics & Statistics, vol 298. Springer, Singapore. https://doi.org/10.1007/978-981-15-0294-1_10

Download citation

Publish with us

Policies and ethics

Search

Navigation