Inventiones mathematicae

, Volume 208, Issue 1, pp 109–189 | Cite as

The Z-invariant massive Laplacian on isoradial graphs

  • Cédric Boutillier
  • Béatrice de TilièreEmail author
  • Kilian Raschel


We introduce a one-parameter family of massive Laplacian operators \((\Delta ^{m(k)})_{k\in [0,1)}\) defined on isoradial graphs, involving elliptic functions. We prove an explicit formula for the inverse of \(\Delta ^{m(k)}\), the massive Green function, which has the remarkable property of only depending on the local geometry of the graph, and compute its asymptotics. We study the corresponding statistical mechanics model of random rooted spanning forests. We prove an explicit local formula for an infinite volume Boltzmann measure, and for the free energy of the model. We show that the model undergoes a second order phase transition at \(k=0\), thus proving that spanning trees corresponding to the Laplacian introduced by Kenyon (Invent Math 150(2):409–439, 2002) are critical. We prove that the massive Laplacian operators \((\Delta ^{m(k)})_{k\in (0,1)}\) provide a one-parameter family of Z-invariant rooted spanning forest models. When the isoradial graph is moreover \({\mathbb {Z}}^2\)-periodic, we consider the spectral curve of the massive Laplacian. We provide an explicit parametrization of the curve and prove that it is Harnack and has genus 1. We further show that every Harnack curve of genus 1 with \((z,w)\leftrightarrow (z^{-1},w^{-1})\) symmetry arises from such a massive Laplacian.



We warmly thank Erwan Brugallé for very helpful discussions on Harnack curves. We acknowledge support from the Agence Nationale de la Recherche (projet MAC2: ANR-10-BLAN-0123) and from the Région Centre-Val de Loire (projet MADACA). We thank the referee for her/his useful comments and suggestions which led us to improve the presentation of this paper. We also thank her/him for the arguments allowing to greatly simplify the proof of Proposition 6 and of Theorem 41.


  1. 1.
    Adler, V.E., Bobenko, A.I., Suris, Y.B.: Classification of integrable equations on quad-graphs. The consistency approach. Commun. Math. Phys. 233(3), 513–543 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arbarello, E., Cornalba, M., Griffiths, P.A., Harris, J.: Geometry of algebraic curves. Vol. I, volume 267 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, New York (1985)Google Scholar
  3. 3.
    Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55. For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. (1964)Google Scholar
  4. 4.
    Au-Yang, H., Perk, J.H.H.: Correlation functions and susceptibility in the Z-invariant Ising model. In: Kashiwara, M., Miwa, T. (eds.) MathPhys Odyssey 2001, Progress in Mathematical Physics, vol. 23, pp. 23–48. Birkhäuser, Boston (2002)Google Scholar
  5. 5.
    Baxter, R.J.: Solvable eight-vertex model on an arbitrary planar lattice. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 289(1359), 315–346 (1978)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Baxter, R.J.: Free-fermion, checkerboard and \({Z}\)-invariant lattice models in statistical mechanics. Proc. R. Soc. Lond. Ser. A 404(1826), 1–33 (1986)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Baxter, R.J.: Exactly solved models in statistical mechanics. Academic Press, London (1989) [Reprint of the 1982 original (1989)]Google Scholar
  8. 8.
    Boutillier, C., de Tilière, B.: The critical \(Z\)-invariant Ising model via dimers: the periodic case. Probab. Theory Related Fields 147, 379–413 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Boutillier, C., de Tilière, B.: The critical \(Z\)-invariant Ising model via dimers: locality property. Commun. Math. Phys. 301(2), 473–516 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bodini, O., Fernique, T., Rémila, É.: A characterization of flip-accessibility for rhombus tilings of the whole plane. Inf. Comput. 206(9–10), 1065–1073 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Benjamini, I., Lyons, R., Peres, Y., Schramm, O.: Uniform spanning forests. Ann. Probab. 29(1), 1–65 (2001)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Bobenko, A.I., Mercat, C., Suris, Y.B.: Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green’s function. J. Reine Angew. Math. 583, 117–161 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Burton, R., Pemantle, R.: Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances. Ann. Probab. 21(3), 1329–1371 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Brugallé, E.: Pseudoholomorphic simple Harnack curves. Enseign. Math. 61(3/4), 483–498 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Bobenko, A.I., Suris, Y.B.: Discrete differential geometry, Graduate Studies in Mathematics, vol. 98. . Integrable structure. American Mathematical Society, Providence (2008)Google Scholar
  16. 16.
    Cimasoni, D., Duminil-Copin, H.: The critical temperature for the Ising model on planar doubly periodic graphs. Electron. J. Probab. 18(44), 1–18 (2013)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Copson, E.T.: Asymptotic expansions. Cambridge Tracts in Mathematics and Mathematical Physics, vol. 55. Cambridge University Press, New York (1965)Google Scholar
  18. 18.
    Cook, R.J., Thomas, A.D.: Line bundles and homogeneous matrices. Q. J. Math. 30(4), 423–429 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    de Bruijn, N.G.: Algebraic theory of Penrose’s non-periodic tilings of the plane. I. Indag. Math. (Proc.) 84(1), 39–52 (1981)CrossRefzbMATHGoogle Scholar
  20. 20.
    de Bruijn, N.G.: Algebraic theory of Penrose’s non-periodic tilings of the plane. II. Indag. Math. (Proc.) 84(1), 53–66 (1981)CrossRefzbMATHGoogle Scholar
  21. 21.
    de Tilière, B.: Quadri-tilings of the plane. Probab. Theory Related Fields 137(3–4), 487–518 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Duffin, R.J.: Potential theory on a rhombic lattice. J. Combin. Theory 5, 258–272 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Gel’fand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, resultants, and multidimensional determinants. Mathematics: Theory and Applications. Birkhäuser, Boston (1994)Google Scholar
  24. 24.
    Kennelly, A.E.: The equivalence of triangles and three-pointed stars in conducting networks. Electr. World Eng. 34, 413–414 (1899)Google Scholar
  25. 25.
    Kenyon, R.: Tiling a polygon with parallelograms. Algorithmica 9(4), 382–397 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kenyon, R.: The Laplacian and Dirac operators on critical planar graphs. Invent. Math. 150(2), 409–439 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kenyon, R.: An introduction to the dimer model. In: School and Conference on Probability Theory, ICTP Lect. Notes, XVII, pp. 267–304 (electronic). Abdus Salam Int. Cent. Theoret. Phys., Trieste (2004)Google Scholar
  28. 28.
    Kirchhoff, G.: Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Ströme geführt wird. Ann. Phys. 148, 497–508 (1847)CrossRefGoogle Scholar
  29. 29.
    Kenyon, R., Okounkov, A.: Planar dimers and Harnack curves. Duke Math. J. 131(3), 499–524 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Kenyon, R., Schlenker, J.-M.: Rhombic embeddings of planar quad-graphs. Trans. Am. Math. Soc. 357(9), 3443–3458 (2005). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Lawden, D.F.: Elliptic functions and applications. Applied Mathematical Sciences, vol. 80. Springer, New York (1989)Google Scholar
  32. 32.
    Li, Z.: Critical temperature of periodic Ising models. Commun. Math. Phys. 315, 337–381 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Lis, M.: Phase transition free regions in the Ising model via the Kac–Ward operator. Commun. Math. Phys. 331(3), 1071–1086 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Mercat, C.: Discrete Riemann surfaces and the Ising model. Commun. Math. Phys. 218(1), 177–216 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Mercat, C.: Exponentials form a basis of discrete holomorphic functions on a compact. Bull. Soc. Math. Fr. 132(2), 305–326 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Mikhalkin, G.: Real algebraic curves, the moment map and amoebas. Ann. of Math. 151(1), 309–326 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Mikhalkin, G., Rullgård, H.: Amoebas of maximal area. Int. Math. Res. Notices 2001(9), 441 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Natanzon, S.M.: Klein surfaces. Russ. Math. Surveys 45(6), 53–108 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Onsager, L.: Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. 65(3–4), 117–149 (1944)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Perk, J.H.H., Au-Yang, H.: Yang–Baxter equations. In: Françoise, J.-P., Naber, G.L., Tsun, T.S. (eds.) Encyclopedia of Mathematical Physics, pp. 465–473. Academic Press, Oxford (2006)CrossRefGoogle Scholar
  41. 41.
    Pemantle, R., Wilson, M.C.: Analytic combinatorics in several variables, Cambridge Studies in Advanced Mathematics, vol. 140. Cambridge University Press, Cambridge (2013)Google Scholar
  42. 42.
    Thurston, W.P.: Conway’s tiling groups. Am. Math. Mon. 97(8), 757–773 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Viro, O.: What is an amoeba? Notices AMS 49(8), 916–917 (2002)Google Scholar
  44. 44.
    Wilson, D.B.: Generating random spanning trees more quickly than the cover time. In: Proceedings of the Twenty-eighth Annual ACM Symposium on the Theory of Computing (Philadelphia, PA, 1996), pp. 296–303. ACM, New York (1996)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Cédric Boutillier
    • 1
  • Béatrice de Tilière
    • 2
    Email author
  • Kilian Raschel
    • 3
  1. 1.Laboratoire de Probabilités et Modèles AléatoiresUniversité Pierre et Marie CurieParisFrance
  2. 2.Laboratoire d’Analyse et de Mathématiques AppliquéesUniversité Paris-Est CréteilCréteilFrance
  3. 3.CNRS, Laboratoire de Mathématiques et Physique ThéoriqueUniversité de ToursToursFrance

Personalised recommendations