Height representation of XOR-Ising loops via bipartite dimers

Cédric Boutillier (UPMC)
Béatrice de Tilière (UPMC)

Abstract


The XOR-Ising model on a graph consists of random spin configurations on vertices of the graph obtained by taking the product at each vertex of the spins of two independent Ising models. In this paper, we explicitly relate loop configurations of the XOR-Ising model and those of a dimer model living on a decorated, bipartite version of the Ising graph. This result is proved for graphs embedded in compact surfaces of genus $g$. Using this fact, we then prove that XOR-Ising loops have the same law as level lines of the height function of this bipartite dimer model. At criticality, the height function is known to converge weakly in distribution to $\frac{1}{\sqrt{\pi}}$ a Gaussian free field. As a consequence, results of this paper shed a light on the occurrence of the Gaussian free field in the XOR-Ising model. In particular, they provide a step forward in the solution of Wilson's conjecture, stating that the scaling limit of XOR-Ising loops are level lines of the Gaussian free field.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-33

Publication Date: September 4, 2014

DOI: 10.1214/EJP.v19-2449

References

  • J. Ashkin and E. Teller. Statistics of two-dimensional lattices with four components. Phys. Rev., 64:178--184, Sep 1943.
  • Boutillier, Cédric; de Tilière, Béatrice. The critical ${\bf Z}$-invariant Ising model via dimers: the periodic case. Probab. Theory Related Fields 147 (2010), no. 3-4, 379--413. MR2639710
  • Boutillier, Cédric; de Tilière, Béatrice. The critical $Z$-invariant Ising model via dimers: locality property. Comm. Math. Phys. 301 (2011), no. 2, 473--516. MR2764995
  • D. Cimasoni and H. Duminil-Copin. The critical temperature for the Ising model on planar doubly periodic graphs. ArXiv e-prints, September 2012.
  • D. Chelkak, C. Hongler, and K. Izyurov. Conformal Invariance of Spin Correlations in the Planar Ising Model. ArXiv e-prints, February 2012.
  • Cimasoni, David; Reshetikhin, Nicolai. Dimers on surface graphs and spin structures. I. Comm. Math. Phys. 275 (2007), no. 1, 187--208. MR2335773
  • Cimasoni, David; Reshetikhin, Nicolai. Dimers on surface graphs and spin structures. II. Comm. Math. Phys. 281 (2008), no. 2, 445--468. MR2410902
  • D. Chelkak and S. Smirnov. Universality in the 2D Ising model and conformal invariance of fermionic observables. Invent. Math., pages 1--66, 2009.
  • de Tilière, Béatrice. Quadri-tilings of the plane. Probab. Theory Related Fields 137 (2007), no. 3-4, 487--518. MR2278466
  • B. de~Tilière. Scaling limit of isoradial dimer models and the case of triangular quadri-tilings. Ann. Inst. H. Poincaré Probab. Statist., 43(6):729--750, 2007.
  • J. Dubédat. Dimers and analytic torsion I. ArXiv e-prints, October 2011.
  • J. Dubédat. Exact bosonization of the Ising model. ArXiv e-prints, December 2011.
  • Dolbilin, N. P.; Zinovʹev, Yu. M.; Mishchenko, A. S.; Shtanʹko, M. A.; Shtogrin, M. I. Homological properties of two-dimensional coverings of lattices on surfaces. (Russian) Funktsional. Anal. i Prilozhen. 30 (1996), no. 3, 19--33, 95; translation in Funct. Anal. Appl. 30 (1996), no. 3, 163--173 (1997) MR1435135
  • C. Fan. On critical properties of the ashkin-teller model. Phys. Lett. A, 39(2):136, 1972.
  • M. E. Fisher. On the dimer solution of planar Ising models. J. Math. Phys., 7:1776--1781, October 1966.
  • Fulton, William. Algebraic topology. A first course. Graduate Texts in Mathematics, 153. Springer-Verlag, New York, 1995. xviii+430 pp. ISBN: 0-387-94326-9; 0-387-94327-7 MR1343250
  • C. Fan and F. Y. Wu. General lattice model of phase transitions. Phys. Rev. B, 2:723--733, Aug 1970.
  • Galluccio, Anna; Loebl, Martin. On the theory of Pfaffian orientations. I. Perfect matchings and permanents. Electron. J. Combin. 6 (1999), Research Paper 6, 18 pp. (electronic). MR1670286
  • Ikhlef, Yacine; Rajabpour, Mohammad Ali. Discrete holomorphic parafermions in the Ashkin-Teller model and SLE. J. Phys. A 44 (2011), no. 4, 042001, 11 pp. MR2754707
  • P. W. Kasteleyn. The statistics of dimers on a lattice: I. the number of dimer arrangements on a quadratic lattice. Physica, 27:1209--1225, December 1961.
  • Kasteleyn, P. W. Graph theory and crystal physics. 1967 Graph Theory and Theoretical Physics pp. 43--110 Academic Press, London MR0253689
  • L. Kadanoff and A. C. Brown. Correlation functions on the critical lines of the Baxter and Ashkin-Teller models. Ann. Phys., 121(1–2):318--342, 1979.
  • Kadanoff, Leo P.; Ceva, Horacio. Determination of an operator algebra for the two-dimensional Ising model. Phys. Rev. B (3) 3 (1971), 3918--3939. MR0389111
  • Kenyon, R. The Laplacian and Dirac operators on critical planar graphs. Invent. Math. 150 (2002), no. 2, 409--439. MR1933589
  • Kenyon, Richard; Okounkov, Andrei; Sheffield, Scott. Dimers and amoebae. Ann. of Math. (2) 163 (2006), no. 3, 1019--1056. MR2215138
  • Kenyon, Richard; Schlenker, Jean-Marc. Rhombic embeddings of planar quad-graphs. Trans. Amer. Math. Soc. 357 (2005), no. 9, 3443--3458 (electronic). MR2146632
  • Kramers, H. A.; Wannier, G. H. Statistics of the two-dimensional ferromagnet. I. Phys. Rev. (2) 60, (1941). 252--262. MR0004803
  • Kramers, H. A.; Wannier, G. H. Statistics of the two-dimensional ferromagnet. II. Phys. Rev. (2) 60, (1941). 263--276. MR0004804
  • M. Kac and J. C. Ward. A combinatorial solution of the two-dimensional Ising model. Phys. Rev., 88:1332--1337, Dec 1952.
  • L. P. Kadanoff and F. J. Wegner. Some critical properties of the eight-vertex model. Phys. Rev. B, 4:3989--3993, Dec 1971.
  • Li, Zhi Bing; Guo, Shuo Hong. Duality for the Ising model on a random lattice and topologic excitons. Nuclear Phys. B 413 (1994), no. 3, 723--734. MR1262557
  • Z. Li. Spectral Curve of Periodic Fisher Graphs. ArXiv e-prints, August 2010.
  • Li, Zhongyang. Critical Temperature of Periodic Ising Models. Comm. Math. Phys. 315 (2012), no. 2, 337--381. MR2971729
  • E. H. Lieb. Residual entropy of square ice. Phys. Rev., 162:162--172, Oct 1967.
  • Massey, William S. A basic course in algebraic topology. Graduate Texts in Mathematics, 127. Springer-Verlag, New York, 1991. xvi+428 pp. ISBN: 0-387-97430-X MR1095046
  • Maunder, C. R. F. Algebraic topology. Reprint of the 1980 edition. Dover Publications, Inc., Mineola, NY, 1996. viii+375 pp. ISBN: 0-486-69131-4 MR1402473
  • Nienhuis, Bernard. Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas. J. Statist. Phys. 34 (1984), no. 5-6, 731--761. MR0751711
  • Percus, Jerome K. One more technique for the dimer problem. J. Mathematical Phys. 10 1969 1881--1888. MR0250899
  • M. Picco and R. Santachiara. Critical interfaces and duality in the Ashkin-Teller model. Phys. Rev. E, 83:061124, Jun 2011.
  • Saleur, H. Partition functions of the two-dimensional Ashkin-Teller model on the critical line. J. Phys. A 20 (1987), no. 16, L1127--L1133. MR0924717
  • Schramm, Oded; Sheffield, Scott. Contour lines of the two-dimensional discrete Gaussian free field. Acta Math. 202 (2009), no. 1, 21--137. MR2486487
  • B. Sutherland. Two-dimensional hydrogen bonded crystals without the ice rule. J. Math. Phys., 11(11):3183--3186, 1970.
  • Tesler, Glenn. Matchings in graphs on non-orientable surfaces. J. Combin. Theory Ser. B 78 (2000), no. 2, 198--231. MR1750896
  • G. H. Wannier. The statistical problem in cooperative phenomena. Rev. Mod. Phys., 17(1):50--60, Jan 1945.
  • F. J. Wegner. Duality relation between the Ashkin-Teller and the eight-vertex model. J. of Phys. C: Solid State Physics, 5(11):L131, 1972.
  • D. B. Wilson. XOR-Ising loops and the Gaussian free field. ArXiv e-prints, February 2011.
  • F. Y. Wu and K. Y. Lin. Staggered ice-rule vertex model -- the Pfaffian solution. Phys. Rev. B, 12:419--428, Jul 1975.
  • F. W. Wu. Ising model with four-spin interactions. Phys. Rev. B, 4:2312--2314, Oct 1971.
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.