Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Critical Ising model and spanning trees partition functions

Béatrice de Tilière

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Abstract

We prove that the squared partition function of the two-dimensional critical Ising model defined on a finite, isoradial graph $\mathsf{G}=(\mathsf{V},\mathsf{E})$, is equal to $2^{|\mathsf{V}|}$ times the partition function of spanning trees of the graph $\bar{\mathsf{G}}$, where $\bar{\mathsf{G}}$ is the graph $\mathsf{G}$ extended along the boundary; edges of $\mathsf{G}$ are assigned Kenyon’s (Invent. Math. 150 (2) (2002) 409–439) critical weights, and boundary edges of $\bar{\mathsf{G}}$ have specific weights. The proof is an explicit construction, providing a new relation on the level of configurations between two classical, critical models of statistical mechanics.

Résumé

Nous montrons que le carré de la fonction de partition du modèle d’Ising critique en dimension deux, défini sur un graphe isoradial $\mathsf{G}=(\mathsf{V},\mathsf{E})$ fini, est égale à $2^{|\mathsf{V}|}$ fois la fonction de partition des arbres couvrants du graphe $\bar{\mathsf{G}}$, où le graphe $\bar{\mathsf{G}}$ est le graphe $\mathsf{G}$ prolongé le long du bord; les arêtes de $\mathsf{G}$ sont munies des poids critiques de Kenyon (Invent. Math. 150 (2) (2002) 409–439), et les arêtes du bord de $\bar{\mathsf{G}}$ ont des poids spécifiques. La preuve consiste en une construction explicite, qui donne une nouvelle relation, au niveau des configurations, entre deux modèles classiques de mécanique statistique au point critique.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 3 (2016), 1382-1405.

Dates
Received: 20 January 2014
Revised: 23 February 2015
Accepted: 8 April 2015
First available in Project Euclid: 28 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1469723524

Digital Object Identifier
doi:10.1214/15-AIHP680

Mathematical Reviews number (MathSciNet)
MR3531713

Zentralblatt MATH identifier
1354.82013

Subjects
Primary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82B27: Critical phenomena 05A19: Combinatorial identities, bijective combinatorics

Keywords
Critical two-dimensional Ising model Critical spanning trees Isoradial graphs Partition functions

Citation

de Tilière, Béatrice. Critical Ising model and spanning trees partition functions. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 3, 1382--1405. doi:10.1214/15-AIHP680. https://projecteuclid.org/euclid.aihp/1469723524


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References

  • [1] R. J. Baxter. Free-fermion, checkerboard and ${Z}$-invariant lattice models in statistical mechanics. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 404 (1826) (1986) 1–33.
  • [2] C. Boutillier and B. de Tilière. The critical $Z$-invariant Ising model via dimers: The periodic case. Probab. Theory Related Fields 147 (2010) 379–413.
  • [3] C. Boutillier and B. de Tilière. Height representation of XOR-Ising loops via bipartite dimers. Electron. J. Probab. 19 (80) (2014) 1–33.
  • [4] D. Cimasoni and H. Duminil-Copin. The critical temperature for the Ising model on planar doubly periodic graphs. Electron. J. Probab. 18 (44) (2013) 1–18.
  • [5] S. Chaiken. A combinatorial proof of the all minors matrix tree theorem. SIAM J. Algebr. Discrete Methods 3 (3) (1982) 319–329.
  • [6] D. Cimasoni. The critical Ising model via Kac–Ward matrices. Comm. Math. Phys. 316 (1) (2012) 99–126.
  • [7] B. de Tilière. From cycle rooted spanning forests to the critical Ising model: An explicit construction. Comm. Math. Phys. 319 (1) (2013) 69–110.
  • [8] J. Dubédat. Exact bosonization of the Ising model, 2011. Available at arXiv:1112.4399.
  • [9] R. J. Duffin. Potential theory on a rhombic lattice. J. Combinatorial Theory 5 (1968) 258–272.
  • [10] M. E. Fisher. On the dimer solution of planar Ising models. J. Math. Phys. 7 (1966) 1776–1781.
  • [11] C. Fan and F. Y. Wu. General lattice model of phase transitions. Phys. Rev. B 2 (1970) 723–733.
  • [12] P. W. Kasteleyn. Graph theory and crystal physics. In Graph Theory and Theoretical Physics 43–110. Academic Press, London, 1967.
  • [13] R. Kenyon. The Laplacian and Dirac operators on critical planar graphs. Invent. Math. 150 (2) (2002) 409–439.
  • [14] G. Kirchhoff. Ueber die auflösung der gleichungen, auf welche man bei der untersuchung der linearen vertheilung galvanischer ströme geführt wird. Annalen der Physik 148 (1847) 497–508.
  • [15] R. W. Kenyon, J. G. Propp and D. B. Wilson. Trees and matchings. Electron. J. Combin. 7 (1) (2000) R25.
  • [16] G. Kuperberg. An exploration of the permanent-determinant method. Electron. J. Combin. 5 (1) (1998) R46.
  • [17] H. A. Kramers and G. H. Wannier. Statistics of the two-dimensional ferromagnet. Part I. Phys. Rev. 60 (3) (1941) 252–262.
  • [18] H. A. Kramers and G. H. Wannier. Statistics of the two-dimensional ferromagnet. Part II. Phys. Rev. 60 (3) (1941) 263–276.
  • [19] M. Kac and J. C. Ward. A combinatorial solution of the two-dimensional Ising model. Phys. Rev. II. Ser. 88 (1952) 1332–1337.
  • [20] L. P. Kadanoff and F. J. Wegner. Some critical properties of the eight-vertex model. Phys. Rev. B 4 (1971) 3989–3993.
  • [21] Z. Li. Critical temperature of periodic Ising models. Comm. Math. Phys. 315 (2012) 337–381.
  • [22] M. Lis. Phase transition free regions in the Ising model via the Kac–Ward operator. Comm. Math. Phys. 331 (2014) 1071–1086.
  • [23] C. Mercat. Discrete Riemann surfaces and the Ising model. Comm. Math. Phys. 218 (1) (2001) 177–216.
  • [24] B. Nienhuis. Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas. J. Stat. Phys. 34 (5–6) (1984) 731–761.
  • [25] H. N. V. Temperley. In Combinatorics: Proceedings of the British Combinatorial Conference 1973 202–204. London Mathematical Society Lecture Notes Series 13. Cambridge University Press, Cambridge, 1974. Contributed talk by H. N. V. Temperley.
  • [26] H. N. V. Temperley and M. E. Fisher. Dimer problem in statistical mechanics – An exact result. Philos. Mag. (8) 6 (1961) 1061–1063.
  • [27] W. T. Tutte. The dissection of equilateral triangles into equilateral triangles. Proc. Cambridge Philos. Soc. 44 (1948) 463–482.
  • [28] F. Y. Wu and K. Y. Lin. Staggered ice-rule vertex model – The Pfaffian solution. Phys. Rev. B 12 (1975) 419–428.
  • [29] F. W. Wu. Ising model with four-spin interactions. Phys. Rev. B 4 (1971) 2312–2314.