Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 079, 90 pages      arXiv:2405.03259      https://doi.org/10.3842/SIGMA.2025.079

The Ising Model Coupled to 2D Gravity: Genus Zero Partition Function

Maurice Duits a, Nathan Hayford a and Seung-Yeop Lee b
a) Department of Mathematics, Royal Institute of Technology (KTH), Stockholm, Sweden
b) Department of Mathematics and Statistics, University of South Florida, Tampa, FL, USA

Received January 31, 2025, in final form September 03, 2025; Published online September 24, 2025

Abstract
We compute the genus 0 free energy for the 2-matrix model with quartic interactions, which acts as a generating function for the Ising model's partition function on a random, 4-regular, planar graph. This is consistent with the predictions of Kazakov and Boulatov on this model, as well as subsequent confirmation of this formula using combinatorial methods. We also provide a new parametric formula for the free energy and give a characterization of the phase space. Our analysis is based on a steepest descent Riemann-Hilbert analysis of the associated biorthogonal polynomials and the corresponding isomonodromic $\tau$-function. A key ingredient in the analysis is a parametrization of the spectral curve. This analysis lays the groundwork for the subsequent study of the multicritical point, which we will study in a forthcoming work.

Key words: 2-matrix model; Riemann-Hilbert analysis; asymptotic analysis; graphical enumeration; Ising model.

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References

  1. Albenque M., Ménard L., Geometric properties of spin clusters in random triangulations coupled with an Ising model, arXiv:2201.11922.
  2. Albenque M., Ménard L., Schaeffer G., Local convergence of large random triangulations coupled with an Ising model, Trans. Amer. Math. Soc. 374 (2021), 175-217, arXiv:1812.03140.
  3. Ambjørn J., Anagnostopoulos K.N., Magnea U., Singularities of the partition function for the Ising model coupled to $2$D quantum gravity, Modern Phys. Lett. A 12 (1997), 1605-1627, arXiv:hep-lat/9705004.
  4. Ambjørn J., Anagnostopoulos K.N., Magnea U., Complex zeros of the $2$D Ising model on dynamical random lattices, Nuclear Phys. B 63 (1998), 751-753, arXiv:hep-lat/9708014.
  5. Aptekarev A.I., Multiple orthogonal polynomials, J. Comput. Appl. Math. 99 (1998), 423-447.
  6. Baxter R.J., Exactly solved models in statistical mechanics, Academic Press, London, 1989.
  7. Belavin A.A., Polyakov A.M., Zamolodchikov A.B., Infinite conformal symmetry in two-dimensional quantum field theory, Nuclear Phys. B 241 (1984), 333-380.
  8. Bernardi O., Bousquet-Mélou M., Counting colored planar maps: algebraicity results, J. Combin. Theory Ser. B 101 (2011), 315-377, arXiv:0909.1695.
  9. Bertola M., Marchal O., The partition function of the two-matrix model as an isomonodromic $\tau$ function, J. Math. Phys. 50 (2009), 013529, 17 pages, arXiv:0809.1598.
  10. Bertola M., Tovbis A., Asymptotics of orthogonal polynomials with complex varying quartic weight: global structure, critical point behavior and the first Painlevé equation, Constr. Approx. 41 (2015), 529-587, arXiv:1108.0321.
  11. Bessis D., Itzykson C., Zuber J.-B., Quantum field theory techniques in graphical enumeration, Adv. in Appl. Math. 1 (1980), 109-157.
  12. Bleher P., Deaño A., Painlevé I double scaling limit in the cubic random matrix model, Random Matrices Theory Appl. 5 (2016), 1650004, 58 pages, arXiv:1310.3768.
  13. Bleher P., Gharakhloo R., McLaughlin K.T.-R., Phase diagram and topological expansion in the complex quartic random matrix model, Comm. Pure Appl. Math. 77 (2024), 1405-1485, arXiv:2112.09412.
  14. Boulatov D.V., Kazakov V.A., The Ising model on a random planar lattice: the structure of the phase transition and the exact critical exponents, Phys. Lett. B 186 (1987), 379-384.
  15. Boulatov D.V., Kazakov V.A., Kostov I.K., Migdal A.A., Analytical and numerical study of a model of dynamically triangulated random surfaces, Nuclear Phys. B 275 (1986), 641-686.
  16. Bousquet-Mélou M., Schaeffer G., The degree distribution in bipartite planar maps: applications to the Ising model, arXiv:math.CO/0211070.
  17. Bouttier J., Di Francesco P., Guitter E., Combinatorics of bicubic maps with hard particles, J. Phys. A 38 (2005), 4529-4559, arXiv:math.CO/0501344.
  18. Bouttier J., Di Francesco P., Guitter E., Blocked edges on Eulerian maps and mobiles: application to spanning trees, hard particles and the Ising model, J. Phys. A 40 (2007), 7411-7440, arXiv:math.CO/0702097.
  19. Brézin É., Douglas M.R., Kazakov V., Shenker S.H., The Ising model coupled to $2$D gravity. A nonperturbative analysis, Phys. Lett. B 237 (1990), 43-46.
  20. Brézin E., Itzykson C., Parisi G., Zuber J.-B., Planar diagrams, Comm. Math. Phys. 59 (1978), 35-51.
  21. Brézin É., Kazakov V.A., Exactly solvable field theories of closed strings, Phys. Lett. B 236 (1990), 144-150.
  22. Cardy J.L., Conformal invariance and the Yang-Lee edge singularity in two dimensions, Phys. Rev. Lett. 54 (1985), 1354-1356.
  23. Chen L., Turunen J., Critical Ising model on random triangulations of the disk: enumeration and local limits, Comm. Math. Phys. 374 (2020), 1577-1643, arXiv:1806.06668.
  24. Chen L., Turunen J., Ising model on random triangulations of the disk: phase transition, Comm. Math. Phys. 397 (2023), 793-873, arXiv:2003.09343.
  25. Chester C., Friedman B., Ursell F., An extension of the method of steepest descents, Proc. Cambridge Philos. Soc. 53 (1957), 599-611.
  26. Claeys T., Vanlessen M., Universality of a double scaling limit near singular edge points in random matrix models, Comm. Math. Phys. 273 (2007), 499-532, arXiv:math-ph/0607043.
  27. Crnković v., Ginsparg P., Moore G., The Ising model, the Yang-Lee edge singularity, and $2$D quantum gravity, Phys. Lett. B 237 (1990), 196-201.
  28. David F., A model of random surfaces with nontrivial critical behaviour, Nuclear Phys. B 257 (1985), 543-576.
  29. de Albuquerque L.C., Alves N.A., Dalmazi D., Yang-Lee zeros of the Ising model on random graphs of non planar topology, Nuclear Phys. B 580 (2000), 739-756, arXiv:hep-th/9912270.
  30. Deift P., Kriecherbauer T., McLaughlin K.T.-R., Venakides S., Zhou X., Strong asymptotics of orthogonal polynomials with respect to exponential weights, Comm. Pure Appl. Math. 52 (1999), 1491-1552.
  31. Deift P., Zhou X., A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. 137 (1993), 295-368, arXiv:math.AP/9201261.
  32. Di Francesco P., Ginsparg P., Zinn-Justin J., $2$D gravity and random matrices, Phys. Rep. 254 (1995), 133, arXiv:hep-th/9306153.
  33. Douglas M.R., Strings in less than one dimension and the generalized KdV hierarchies, Phys. Lett. B 238 (1990), 176-180.
  34. Douglas M.R., Shenker S.H., Strings in less than one dimension, Nuclear Phys. B 335 (1990), 635-654.
  35. Duits M., Geudens D., A critical phenomenon in the two-matrix model in the quartic/quadratic case, Duke Math. J. 162 (2013), 1383-1462, arXiv:1111.2162.
  36. Duits M., Geudens D., Kuijlaars A.B.J., A vector equilibrium problem for the two-matrix model in the quartic/quadratic case, Nonlinearity 24 (2011), 951-993, arXiv:1007.3137.
  37. Duits M., Hayford N., The Ising model coupled to $2$D gravity: critical partition function, in preparation.
  38. Duits M., Kuijlaars A.B.J., Painlevé I asymptotics for orthogonal polynomials with respect to a varying quartic weight, Nonlinearity 19 (2006), 2211-2245, arXiv:math.CA/0605201.
  39. Duits M., Kuijlaars A.B.J., Universality in the two-matrix model: a Riemann-Hilbert steepest-descent analysis, Comm. Pure Appl. Math. 62 (2009), 1076-1153, arXiv:0807.4814.
  40. Duits M., Kuijlaars A.B.J., Mo M.Y., The Hermitian two matrix model with an even quartic potential, Mem. Amer. Math. Soc. 217 (2012), v+105 pages, arXiv:1010.4282.
  41. Eynard B., Large-$N$ expansion of the 2-matrix model, J. High Energy Phys. 2003 (2003), no. 1, 051, 38 pages, arXiv:hep-th/0210047.
  42. Eynard B., Counting surfaces. CRM Aisenstadt chair lectures, Prog. Math. Phys., Vol. 70, Birkhäuser, Cham, 2016.
  43. Fokas A.S., Its A.R., Kitaev A.V., Discrete Painlevé equations and their appearance in quantum gravity, Comm. Math. Phys. 142 (1991), 313-344.
  44. Fokas A.S., Its A.R., Kitaev A.V., The isomonodromy approach to matrix models in $2$D quantum gravity, Comm. Math. Phys. 147 (1992), 395-430.
  45. Fukuma M., Kawai H., Nakayama R., Explicit solution for $p$-$q$ duality in two-dimensional quantum gravity, Comm. Math. Phys. 148 (1992), 101-116.
  46. Ginsparg P., Goulian M., Plesser M.R., Zinn-Justin J., $(p,q)$ string actions, Nuclear Phys. B 342 (1990), 539-563.
  47. Gross D.J., Migdal A.A., A nonperturbative treatment of two-dimensional quantum gravity, Nuclear Phys. B 340 (1990), 333-365.
  48. Gross D.J., Migdal A.A., Nonperturbative two-dimensional quantum gravity, Phys. Rev. Lett. 64 (1990), 127-130.
  49. Hayford N., The Ising model coupled to gravity: higher-order Painlevé equations/the $(3,4)$ string equation, arXiv:2405.03260.
  50. Ising E., Beitrag zur theorie des ferromagnetismus, Z. Physik 31 (1925), 253-258.
  51. Itzykson C., Zuber J.-B., The planar approximation. II, J. Math. Phys. 21 (1980), 411-421.
  52. Joshi N., Kitaev A.V., Treharne P.A., On the linearization of the first and second Painlevé equations, J. Phys. A 42 (2009), 055208, 18 pages, arXiv:0806.0271.
  53. Kadanoff L.P., Scaling laws for ising models near ${T}_c$, Phys. Phys. Fiz. 2 (1966), 263-272.
  54. Kapaev A.A., Quasi-linear stokes phenomenon for the Painlevé first equation, J. Phys. A 37 (2004), 11149-11167, arXiv:nlin.SI/0404026.
  55. Kapaev A.A., Kitaev A.V., Connection formulae for the first Painlevé transcendent in the complex domain, Lett. Math. Phys. 27 (1993), 243-252.
  56. Kazakov V.A., Bilocal regularization of models of random surfaces, Phys. Lett. B 150 (1985), 282-284.
  57. Kazakov V.A., Ising model on a dynamical planar random lattice: exact solution, Phys. Lett. A 119 (1986), 140-144.
  58. Kazakov V.A., Kostov I.K., Migdal A.A., Critical properties of randomly triangulated planar random surfaces, Phys. Lett. B 157 (1985), 295-300.
  59. Knizhnik V.G., Polyakov A.M., Zamolodchikov A.B., Fractal structure of $2$D-quantum gravity, Modern Phys. Lett. A 3 (1988), 819-826.
  60. Krantz S.G., Parks H.R., The implicit function theorem. History, theory, and applications, Mod. Birkhäuser Class., Birkhäuser, New York, 2013.
  61. Kuijlaars A.B.J., McLaughlin K.T.-R., A Riemann-Hilbert problem for biorthogonal polynomials, J. Comput. Appl. Math. 178 (2005), 313-320, arXiv:math.CV/0310204.
  62. Kuijlaars A.B.J., Silva G.L.F., S-curves in polynomial external fields, J. Approx. Theory 191 (2015), 1-37, arXiv:1311.7026.
  63. Lee T.D., Yang C.N., Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model, Phys. Rev. 87 (1952), 410-419.
  64. Liechty K., Wang D., Two lax systems for the Painlevé II equation, and two related kernels in random matrix theory, SIAM J. Math. Anal. 48 (2016), 3618-3666, arXiv:1601.01603.
  65. Martínez-Finkelshtein A., Silva G.L.F., Critical measures for vector energy: global structure of trajectories of quadratic differentials, Adv. Math. 302 (2016), 1137-1232, arXiv:1509.06704.
  66. Martínez-Finkelshtein A., Silva G.L.F., Critical measures for vector energy: asymptotics of non-diagonal multiple orthogonal polynomials for a cubic weight, Adv. Math. 349 (2019), 246-315, arXiv:1805.01748.
  67. Martínez-Finkelshtein A., Silva G.L.F., Spectral curves, variational problems and the Hermitian matrix model with external source, Comm. Math. Phys. 383 (2021), 2163-2242, arXiv:1907.08108.
  68. McCoy B.M., Wu T.T., The two-dimensional Ising model, Harvard University Press, Cambridge, MA, 1973.
  69. Mehta M.L., A method of integration over matrix variables, Comm. Math. Phys. 79 (1981), 327-340.
  70. Mehta M.L., Random matrices, 3rd ed., Pure Appl. Math. (Amsterdam), Vol. 142,Elsevier, Amsterdam, 2004.
  71. Miller P.D., Applied asymptotic analysis, Grad. Stud. Math., Vol. 75, American Mathematical Society, Providence, RI, 2006.
  72. Nikishin E.M., Sorokin V.N., Rational approximations and orthogonality, Transl. Math. Monogr., Vol. 92, American Mathematical Society, Providence, RI, 1991.
  73. Onsager L., Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev. 65 (1944), 117-149.
  74. Orantin N., Chain of matrices, loop equations, and topological recursion, in The Oxford Handbook of Random Matrix Theory, Oxford University Press, Oxford, 2011, 329-352, arXiv:0911.5089.
  75. Polyakov A.M., Lecture at Northeastern University (from the lecture notes of P. Ginsparg and G. Moore), arXiv:hep-th/9304011.
  76. Polyakov A.M., Quantum geometry of bosonic strings, Phys. Lett. B 103 (1981), 207-210.
  77. Rakhmanov E.A., Orthogonal polynomials and $S$-curves, in Recent Advances in Orthogonal Polynomials, Special Functions, and Their Applications, Contemp. Math., Vol. 578, American Mathematical Society, Providence, RI, 2012, 195-239.
  78. Stanley R.P., Enumerative combinatorics. Vol. 2, Cambridge Stud. Adv. Math., Vol. 62, Cambridge University Press, Cambridge, 1999.
  79. Staudacher M., The Yang-Lee edge singularity on a dynamical planar random surface, Nuclear Phys. B 336 (1990), 349-362.
  80. Turunen J., Interfaces in the vertex-decorated Ising model on random triangulations of the disk arXiv:2003.11012.
  81. Van Assche W., Coussement E., Some classical multiple orthogonal polynomials, J. Comput. Appl. Math. 127 (2001), 317-347, arXiv:math.CA/0103131.
  82. Whittaker E.T., Watson G.N., A course of modern analysis, Cambridge Math. Lib., Cambridge University Press, Cambridge, 1996.
  83. Wilson K.G., Renormalization group and critical phenomena. I. Renormalization and the Kadanoff scaling picture, Phys. Rev. B 4 (1971), 3174-3183.
  84. Wilson K.G., Renormalization group and critical phenomena. II. Phase-space cell analysis of critical behavior, Phys. Rev. B 4 (1971), 3184-3205.
  85. Yang C.N., The spontaneous magnetization of a two-dimensional Ising model, Phys. Rev. 85 (1952), 808-816.
  86. Yang C.N., Lee T.D., Statistical theory of equations of state and phase transitions. I. Theory of condensation, Phys. Rev. 87 (1952), 404-409.
  87. Zamolodchikov A., Ishimoto Y., Massive Majorana fermion coupled to two-dimensional gravity and random lattice Ising model, Theoret. and Math. Phys. 147 (2006), 755-776.
  88. Zinn-Justin P., Zuber J.-B., On some integrals over the ${\rm U}(N)$ unitary group and their large $N$ limit, J. Phys. A 36 (2003), 3173-3193, arXiv:math-ph/0209019.

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