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

SIGMA 3 (2007), 099, 43 pages      arXiv:0710.2756      https://doi.org/10.3842/SIGMA.2007.099
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

### From Holonomy of the Ising Model Form Factors to n-Fold Integrals and the Theory of Elliptic Curves

Salah Boukraa a, Saoud Hassani b, Jean-Marie Maillard c and Nadjah Zenine b
a) LPTHIRM and Département d'Aéronautique, Université de Blida, Algeria
b) Centre de Recherche Nucléaire d'Alger, 2 Bd. Frantz Fanon, BP 399, 16000 Alger, Algeria
c) LPTMC, Université de Paris 6, Tour 24, 4ème étage, case 121, 4 Place Jussieu, 75252 Paris Cedex 05, France

Received September 19, 2007, in final form October 07, 2007; Published online October 15, 2007

Abstract
We recall the form factors f(j)N,N corresponding to the l-extension C(N,N; l) of the two-point diagonal correlation function of the Ising model on the square lattice and their associated linear differential equations which exhibit both a "Russian-doll" nesting, and a decomposition of the linear differential operators as a direct sum of operators (equivalent to symmetric powers of the differential operator of the complete elliptic integral E). The scaling limit of these differential operators breaks the direct sum structure but not the "Russian doll" structure, the "scaled" linear differential operators being no longer Fuchsian. We then introduce some multiple integrals of the Ising class expected to have the same singularities as the singularities of the n-particle contributions c(n) to the susceptibility of the square lattice Ising model. We find the Fuchsian linear differential equations satisfied by these multiple integrals for n = 1, 2, 3, 4 and, only modulo a prime, for n = 5 and 6, thus providing a large set of (possible) new singularities of the c(n). We get the location of these singularities by solving the Landau conditions. We discuss the mathematical, as well as physical, interpretation of these new singularities. Among the singularities found, we underline the fact that the quadratic polynomial condition 1 + 3w + 4w2 = 0, that occurs in the linear differential equation of c(3), actually corresponds to the occurrence of complex multiplication for elliptic curves. The interpretation of complex multiplication for elliptic curves as complex fixed points of generators of the exact renormalization group is sketched. The other singularities occurring in our multiple integrals are not related to complex multiplication situations, suggesting a geometric interpretation in terms of more general (motivic) mathematical structures beyond the theory of elliptic curves. The scaling limit of the (lattice off-critical) structures as a confluent limit of regular singularities is discussed in the conclusion.

Key words: form factors; sigma form of Painlevé VI; two-point correlation functions of the lattice Ising model; Fuchsian linear differential equations; complete elliptic integrals; elliptic representation of Painlevé VI; scaling limit of the Ising model; susceptibility of the Ising model; singular behaviour; Fuchsian linear differential equations; apparent singularities; Landau singularities; pinch singularities; modular forms; Landen transformation; isogenies of elliptic curves; complex multiplication; Heegner numbers; moduli space of curves; pointed curves.

pdf (588 kb)   ps (316 kb)   tex (56 kb)

References

1. Adler A., Ramanan S., Moduli of Abelian varieties, Lecture Notes in Mathematics, Vol. 1644, Springer-Verlag, Heidelberg Berlin, 1996.
2. Au-Yang H., Perk J.H.H., Correlation functions and susceptibility in the Z-invariant Ising model, in MathPhys Odyssee 2001: Integrable Models and Beyond, Editors T. Miwa and M. Kashiwara, Birkhäuser, Boston, 2002, 23-48.
3. Au-Yang H., Perk J.H.H., Critical correlations in a Z-invariant inhomogeneous Ising model, Phys. A 144 (1987), 44-104.
4. Au-Yang H., Perk J.H.H., Ising correlations at the critical temperature, Phys. Lett. A 104 (1984), 131-134, see equation (4) on page 3.
5. Au-Yang H., Perk J.H.H., Star-triangle equations and identities in hypergeometric series, Internat. J. Modern Phys. B 16 (2002), 1853-1865.
6. Au-Yang H., Perk J.H.H., Wavevector-dependent susceptibility in aperiodic planar Ising models, in MathPhys Odyssee 2001: Integrable Models and Beyond, Editors T. Miwa and M. Kashiwara, Birkhäuser, Boston, 2002, 1-21.
7. Babelon O., Bernard D., From form factors to correlation functions: the Ising model, Phys. Lett. B 288 (1992), 113-120.
8. Bailey D.H., Borwein J.M., Crandall R.E., Integrals of the Ising class, J. Phys. A: Math. Gen. 39 (2006), 12271-12302.
9. Ballico E., Casnati G., Fontanari C., On the birational geometry of Moduli spaces of pointed curves, math.AG/0701475.
10. Barad G., The fundamental group of the real moduli spaces M0,n. A preliminary report on the topological aspects of some real algebraic varieties, http://www.geocities.com/gbarad2002/group.pdf.
11. Baxter R.J., Exactly solved models in statistical mechanics, Academic Press, London, 1982.
12. Baxter R.J., Solvable eight vertex model on an arbitrary planar lattice, Phil. Trans. R. Soc. London A 289 (1978), 315-346.
13. Baxter R.J., Partition function of the eight-vertex lattice model, Ann. Physics 70 (1972), 193-228.
Baxter R.J., One-dimensional anisotropic Heisenberg chain, Ann. Physics 70 (1972), 323-337.
14. Baxter R.J., Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain. I. Some fundamental eigenvectors, Ann. Physics 76 (1973), 1-24.
Baxter R.J., Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain. II. Equivalence to a generalized ice-type lattice model, Ann. Physics 76 (1973), 25-47.
Baxter R.J., Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain. III. Eigenvectors of the transfer matrix and the Hamiltonian, Ann. Physics 76 (1973), 48-71.
15. Bazhanov V.V., Mangazeev V.V., The eight-vertex model and Painlevé VI, in Special Issue on Painlevé VI, J. Phys. A: Math. Gen. 39 (2006), 12235-12243, hep-th/0602122.
16. Bazhanov V.V., Nienhuis B., Warnaar O., Lattice Ising model in a field: E8 scattering theory, Phys. Lett. B 322 (1994), 198-206, hep-th/9312169.
17. Bertin J., Peters C., Variations de structure de Hodge, Variétés de Calabi-Yau et symétrie miroir, Panorama et Synthèses, Vol. 3, Société Mathématique de France, 1996.
18. Beukers F., A note on the irrationality of x(2) and x(3), Bull. London Math. Soc. 11 (1979), 268-272.
19. Bloch S., Motives associated to graphs, Jpn. J. Math. 2 (2007), 165-196,
available at http://math.bu.edu/people/kayeats/motives/ graph_rept061017.pdf.
20. Boel R.J., Kasteleyn P.W., Correlation-function identities for general Ising models, Phys. A 93 (1978), 503-516.
21. Boel R.J., Kasteleyn P.W., Correlation-function identities and inequalities for Ising models with pair interactions, Comm. Math. Phys. 161 (1978), 191-208.
22. Boos H.E., Korepin V.E., Evaluation of integrals representing correlations in the XXX Heisenberg spin chain, in MathPhys Odyssey 2001, Editors M. Kashiwara and T. Miwa, Birkhäuser, 2002, 65-108.
23. Boos H.E., Gohmann F., Klumper A., Suzuki J., Factorization of multiple integrals representing the density matrix of a finite segment of the Heisenberg spin chain, in The 75th Anniversary of the Bethe Ansatz, J. Stat. Mech. Theory Exp. 2006 (2006), P04001, 13 pages.
24. Boos H.E., Korepin V.E., Quantum spin chains and Riemann zeta function with odd arguments, J. Phys. A: Math. Gen. 34 (2001), 5311-5316, hep-th/0104008.
25. Boukraa S., Hassani S., Maillard J.-M., McCoy B.M., Orrick W.P., Zenine N., Holonomy of the Ising model form factors, J. Phys. A: Math. Theor. 40 (2007), 75-111, math-ph/0609074.
26. Boukraa S., Hassani S., Maillard J.-M., McCoy B.M., Weil J.A., Zenine N., Fuchs versus Painlevé, J. Phys. A: Math. Theor. 40 (2007), 12589-12605, math-ph/0701014.
27. Boukraa S., Hassani S., Maillard J.-M., McCoy B.M., Weil J.A., Zenine N., Painlevé versus Fuchs, J. Phys. A: Math. Gen. 39 (2006), 12245-12263, math-ph/0602010.
28. Boukraa S., Hassani S., Maillard J.-M., McCoy B.M., Zenine N., The diagonal Ising susceptibility, J. Phys. A: Math. Theor. 40 (2007), 8219-8236, math-ph/0703009.
29. Boukraa S., Hassani S., Maillard J.-M., Zenine N., Landau singularities and singularities of holonomic integrals of the Ising class, J. Phys. A: Math. Theor. 40 (2007), 2583-2614, math-ph/0701016.
30. Boukraa S., Hassani S., Maillard J.-M., Zenine N., Singularities of n-fold integrals of the Ising class and the theory of elliptic curves, J. Phys. A: Math. Theor. 40 (2007), 11713-11748, arXiv:0706.3367.
31. Brown F.C.S., Périodes des espaces des modules M0,n et valeurs zêtas multiples. Multiple zeta values and periods of moduli spaces M0,n, CRAS C. R. Acad. Sci. Paris Ser I 342 (2006), 949-954.
32. Buff X., Fehrenbach J., Lochak P., Schnepps L., Vogel P., Espaces de modules des courbes, groupes modulaires et théorie des champs, Panorama et Synthèses, Numéro 7, Société Mathématique de France, 1999.
33. Cecotti S., Vafa C., Ising model and n = 2 supersymmetric theories, Comm. Math. Phys. 157 (1993), 139-178, hep-th/9209085.
34. Cresson J., Fischler S., Rivoal T., Séries hypergéométriques multiples et polyzêtas, math.NT/0609743.
35. Eden R.J., Landshoff P.V., Olive D.I., Polkinghorne J.C., The analytic S-matrix, Cambridge University Press, 1966.
36. Erdeleyi A., Asymptotic series, Dover Publishing Co., New York, 1956, p. 47.
37. Erdeleyi, Bateman manuscript project, higher transcendental functions, McGraw Hill, New York, 1955.
38. Farkas G., Guibney A., The Mori cones of moduli spaces of pointed curves of small genus, Trans. Amer. Math. Soc. 355 (2003), 1183-1199, math.AG/0111268.
39. Felder G., Varchenko A., The elliptic Gamma function and SL(3, Z) ×Z3, Adv. Math. 156 (2000), 44-76, math.QA/9907061.
40. Feverati G., Grinza P., Integrals of motion from TBA and lattice-conformal dictionary, Nuclear Phys. B 702 (2004), 495-515, hep-th/0405110.
41. Fischler S., Groupes de Rhin-Viola et intégrales multiples, J. Théor. Nombres Bordeaux 15 (2003), 479-534.
42. Fischler S., Intégrales de Brown et de Rhin-Viola pour z(3), math.NT/0609799.
43. Fischler S., Irrationalité de valeurs de zêta (d'après Apéry, Rivoal, ...), Séminaire Bourbaki, Exposé, no. 910, 2003, Astérisque 294 (2004), 27-62, math.NT/0303066.
44. Garnier R., Sur les singularités irrégulières des équations différentielles linéaires, J. Math. Pures et Appl. 2 (1919), 99-198.
45. Gerkmann R., Relative rigid cohomology and deformation of hypersurfaces, Int. Math. Res. Pap. IMRP 2007 (2007), no. 1, Art. ID rpm003, 67 pages.
46. Gervais J.-L., Neveu A., Non-standard 2D critical statistical models from Liouville theory, Nuclear Phys. B 257 (1985), 59-76.
47. Glutsuk A.A., Stokes operators via limit monodromy of generic perturbation, J. Dynam. Control Systems 5 (1999), 101-135.
48. Goncharov A.B., Manin Y., Multiple z-motives and moduli spaces M0,n, Comp. Math. 140 (2004), 1-14, math.AG/0204102.
49. Groeneveld J., Boel R.J., Kasteleyn P.W., Correlation function identities for general planar Ising systems, Phys. A 93 (1978), 138-154.
50. Guzzetti D., The elliptic representation of the general Painlevé VI equation, Comm. Pure Appl. Math. 55 (2002), 1280-1363, math.CV/0108073.
51. Hanna M., The modular equations, Proc. London Math. Soc. 28 (1928), 46-52.
52. Hara Y., Jimbo M., Konno H., Odake S., Shiraishi J., Free field approach to the dilute AL models, J. Math. Phys. 40 (1999), 3791-3826.
53. Harris M., Potential automorphy of odd-dimensional symmetric powers of elliptic curves, and applications, in Algebra, Arithmetic and Geometry - Manin Festschrift, Progress in Mathematics, Birkhäuser, to appear.
54. Hassett B., Tschinkel Y., On the effective cone of the moduli space of pointed rational curves, in Topology and Geometry: Commemorating SISTAG, Contemp. Math. 314 (2002), 83-96, math.AG/0110231.
55. Huttner M., Constructible sets of linear differential equations and effective rational approximations of polylogarithmic functions, Israel J. Math. 153 (2006), 1-44.
56. Huttner M., Equations différientielles fuchsiennes; Approximations du dilogarithme, de z(2) et z(3), Publ. IRMA, Lille, 1997.
57. Jaekel M.T., Maillard J.-M., Inverse functional relations and disorder solutions on the Potts models, J. Phys. A: Math. Gen. 17 (1984), 2079-2094.
58. Jaekel M.T., Maillard J.-M., Inverse functional relations on the Potts model, J. Phys. A: Math. Gen. 15 (1982), 2241-2257.
59. Jimbo M., Kedem R., Konno H., Miwa T., Weston R., Difference equations in spin chains with a boundary, Nuclear Phys. B 448 (1995), 429-456, hep-th/9502060.
60. Jimbo M., Miwa T., Studies on holonomic quantum fields. XVII, Proc. Japan Acad. Ser. A Math. Sci. 56 (1980), 405-410, Erratum, Proc. Japan Acad. Ser. A Math. Sci. 57 (1981), 347.
61. Jimbo M., Miwa T., Nakayashiki A., Difference equations for the correlations of the eight vertex model, J. Phys. A: Math. Gen. 26 (1993), 2199-2209, hep-th/9211066.
62. Katz N.M., Introduction aux travaux récents de Dwork, in Proc. Sympos. Pure Math., Vol. XX (State Univ. New York, Stony Brook, New York, 1969), Amer. Math. Soc., Providence, R.I. 1971, 65-75.
63. Katz N.M., Nilpotent connections and the monodromy theorem: applications of a result of Turrittin, Publications mathématiques de l'IHES 39 (1970), 175-232.
64. Katz N.M., On the differential equations satisfied by period matrices, Inst. Hautes Etudes Sci. Publ. Math. 35 (1968), 223-258.
65. Katz N.M., Rigid local systems, Ann. of Math. Stud., Vol. 139, Princeton University press, 1996.
66. Katz N.M., Travaux de Dwork, in Séminaire Bourbaki (1971/1972), Exp. No. 409, Lecture Notes in Math., Vol. 317, Springer Verlag, 1973, 167-200.
67. Kaufman B., Crystal statistics. II. Partition function evaluated by spinor analysis, Phys. Rev. 76 (1949), 1232-1243.
68. Kaufman B., Onsager L., Short-range order in a binary Ising lattice, Phys. Rev. 76 (1949), 1244-1252.
69. Krattenthaler C., Rivoal T., An identity of Andrews, multiple integrals, and very-well-poised hypergeometric series, Ramanujan J. 13 (2007), 203-219, math.CA/0312148.
70. Kreimer D., Knots and Feynman diagrams, Cambridge Lecture Notes in Physics, Vol. 13, Cambridge University Press, 2000, Chapter 9.
71. Lario J.-C., Elliptic curves with CM defined over extensions of type (2,...,2), available at http://www-ma2.upc.es/~lario/ellipticm.htm.
72. Levin A., Racinet G., Towards multiple elliptic polylogarithms, math.NT/0703237.
73. Lukyanov S., Pugai Y., Multi-point local height probabilities in the integrable RSOS model, Nuclear Phys. B 473 (1996), 631-658, hep-th/9602074.
74. Lyberg I., McCoy B.M., Form factor expansion of the row and diagonal correlation functions of the two dimensional using model, J. Phys. A: Math. Theor. 40 (2007), 3329-3346, math-ph/0612051.
75. Maier R.S., On rationally parametrized modular equations, math.NT/0611041.
76. Maillard J.-M., The inversion relation: some simple examples, J. Physique 46 (1984), 329-341.
77. Maillard J.-M., Boukraa S., Modular invariance in lattice statistical mechanics, Ann. Fond. Louis de Broglie 26 (2001), Special Issue 2, 287-328.
78. Manin Yu.I., Sixth Painlevé equation, Universal elliptic curve, and mirror of P2, Amer. Math. Soc. Transl. Ser. 2 186 (1998), 131-151, alg-geom/9605010.
79. Manojlovic N., Nagy Z., Creation operators and algebraic Bethe ansatz for elliptic quantum group Et,h(so3), J. Phys. A: Math. Theor. 40 (2007), 4181-4191, math.QA/0612087.
80. Martinez J.R., Correlation functions for the Z-invariant Ising model, hep-th/9609135.
81. Mazzocco M., Picard and Chazy solutions to the Painlevé VI equation, Math. Ann. 321 (2001), 157-195, math.AG/9901054.
82. McCoy B.M., Perk J.H.H., Wu T.T., Ising field theory: quadratic difference equations for the n-point Green's functions on the lattice, Phys. Rev. Lett. 46 (1981), 757-760.
83. McCoy B., Tracy C.A., Wu T.T., Painlevé equations of the third kind, J. Math. Phys. 18 (1977), 1058-1092.
84. McCoy B.M., Wu T.T., Nonlinear partial difference equations for the two-dimensional Ising model, Phys. Rev. Lett. 45 (1980), 675-678.
85. Murata M., Sakai H., Yoneda J., Riccati solutions of discrete Painlevé equations with Weyl group symmetry of type E8(1), J. Math. Phys. 44 (2003), 1396-1414, nlin.SI/0210040.
86. Mussardo G., Il Modello di Ising introduzione alla teoria dei campi e delle transizioni di fase, Editor Bollati Boringhieri, 2007.
87. Nickel B., On the singularity structure of the Ising model susceptibility, J. Phys. A: Math. Gen. 32 (1999), 3889-3906.
88. Nickel B., Addendum to `On the singularity structure of the Ising model susceptibility', J. Phys. A: Math. Gen. 33 (2000), 1693-1711.
89. Nickel B., Comment on "The Fuchsian differential equation of the square lattice Ising model c(3) susceptibility" [J. Phys. A: Math. Gen. 37 (2004), 9651-9668] by N. Zenine, S. Boukraa, S. Hassani and J.-M. Maillard, J. Phys. A: Math. Gen. 38 (2005), 4517-4518.
90. Okamoto K., Studies on the Painlevé equations. I. Sixth Painlevé equation P VI, Ann. Mat. Pura Appl. (4) 146 (1987), 337-381.
91. Onsager L., Crystal statistics. I. A two-dimensional model with an order disorder transition, Phys. Rev. 65 (1944), 117-149.
92. Onsager L., Nuovo Cimento 6 (1949), suppl., 261.
93. Orrick W.P., Nickel B.G., Guttmann A.J., Perk J.H.H., The susceptibility of the square lattice Ising model: new developments, J. Statist. Phys. 102 (2001), 795-841, cond-mat/0103074.
94. Palmer J., Tracy C.A., Two-dimensional Ising correlations: convergence of the scaling limit, Adv. in Appl. Math. 2 (1981), 329-388.
95. Pearce P.A., Temperley-Lieb operators and critical A-D-E models, Internat. J. Modern Phys. A 4 (1990), 715-734.
96. Perk J.H.H., Quadratic identities for Ising model correlations, Phys. Lett. A 79 (1980), 3-5.
97. Perk J.H.H., Au-Yang H., Some recent results on pair correlation functions and susceptibilities in exactly solvable models, in Dunk Island Conference in Honor of 60th Birthday of A.J. Guttmann, J. Phys. Conf. Ser. 42 (2006), 231-238, math-ph/0606046.
98. Perk J.H.H., Capel H.W., Time-dependent xx-correlation functions in the one-dimensional XY-model, Phys. A 89 (1977), 265-303, see equation (6.16).
99. Picard E., Mémoire sur la theorie des functions algébriques de deux variables, Journal de Liouville 5 (1889), 135-319.
100. Piezas T. III, Weisstein E.W., j-function, from MathWorld - A Wolfram Web Resource, http://mathworld.wolfram.com/j-Function.html.
101. Racinet G., Doubles mélanges des polylogarithmes multiples aux racines de l'unité, Publ. Math. Inst. Hautes Études Sci. No. 95 (2002), 185-231, math.QA/0202142.
102. Ramis J.-P., Confluence et Résurgence, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (1992), 703-716.
103. Rammal R., Maillard J.-M., q-state Potts model on the checkerboard lattice, J. Phys. A: Math. Gen. 16 (1983), 1073-1081.
104. Rammal R., Maillard J.-M., Some analytical consequences of the inverse relation for the Potts model, J. Phys. A: Math. Gen. 16 (1983), 353-367.
105. Rhoades R.C., Elliptic curves and modular forms (notes based on A Course at the University of Wisconsin - Madison MATH 844 during the Spring 2006 taught by Professor Nigel Boston), available at http://www.math.wisc.edu/~rhoades/Notes/EC.pdf.
106. Sakai H., Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys. 220 (2001), 165-229.
107. Seaton K.A., Batchelor M.T., The dilute A4 models, the E7 mass spectrum and the tricritical Ising model, J. Math. Phys. 43 (2002), 2636-2653, math-ph/0110021.
108. Singer M.F., Testing reducibility of linear differential operators: a group theoretic perspective, Appl. Alg. Eng. Commun. Comp. 7 (1996), no. 2, 77-104.
109. Sorokin V.N., On the measure of transcendency of the number p2, Mat. Sb. 187 (1996), no. 12, 87-120 (English transl.: Sb. Math. 187 (1996), 1819-1852).
110. Sorokin V.N., Apéry's theorem, Vestnik Moskov. Univ. Ser. I Mat. Mekh. (1998), no. 3, 48-53, 74 (English transl.: Moscow Univ. Math. Bull. 53 (1998), no. 3, 48-52).
111. Stanev Y.S., Todorov I., On the Schwartz problem for the su2 Knizhnik-Zamolodchikov equation, Lett. Math. Phys. 35 (1995), 123-134.
112. Sternin B.Yu., Shatalov V.E., On the confluence phenomenon of fuchsian equations, J. Dynam. Control Systems 3 (1997), 433-448.
113. Todorov I.T., Arithmetic features of rational conformal field theory, Ann. Inst. H. Poincaré 63 (1995), 427-453.
114. van der Put M., Singer M.F., Galois theory of linear differential equations, Grundlehren der Mathematischen Wissenschaften, Vol. 328, Springer-Verlag, Berlin, 2003, available at http://www4.ncsu.edu/singer/.
115. van Hoeij M., Rational solutions of the mixed differential equation and its application to factorization of differential operators, in Proceedings ISSAC '96, ACM, New York, 1996, 219-225.
116. Vasilyev D.V., On small linear forms for the values of the Riemann zeta-function at off integers, Doklady NAN Belarusi (Reports of the Belarus National Academy of Sciences) 45 (2001), no. 5, 36-40 (in Russian).
117. Vasilyev D.V., Some formulas for Riemann zeta-function at integer points, Vestnik Moskov. Univ. Ser. I Mat. Mekh. (1996), no. 1, 81-84 (English transl.: Moscow Univ. Math. Bull. 51 (1996), no. 1, 41-43).
118. Warnaar O., Nienhuis B., Seaton K.A., New construction of solvable lattice models including an Ising model in a field, Phys. Rev. Lett. 69 (1992), 710-712.
119. Warnaar O., Pearce P.A., Exceptional structure of the dilute A3 model: E8 and E7 Rogers-Ramanujan identities, J. Phys. A: Math. Gen. 27 (1994), L891-L897, hep-th/9408136.
120. Weisstein E.W., Jacobi theta functions, from MathWorld - A Wolfram Web Resource, http://mathworld.wolfram.com/JacobiThetaFunctions.html.
121. Witte N.S., Isomonodromic deformation theory and the next-to-diagonal correlations of the anisotropic square lattice Ising model, J. Phys. A: Math. Theor. 40 (2007), F491-F501, arXiv:0705.0557.
122. Wu T.T., Theory of Toeplitz determinants and of the spin correlations of the two-dimensional Ising model, Phys. Rev. 149 (1966), 380-440.
123. Wu T.T., McCoy B.M., Tracy C.A., Barouch E., Spin-spin correlation functions for the two dimensional Ising model: exact theory in the scaling region, Phys. Rev. B 13 (1976), 316-374.
124. Yamada K., On the spin-spin correlation function of the Ising square lattice and the zero field susceptibility, Progr. Theoret. Phys. 71 (1984), 1416-1418.
125. Yang C.N., The spontaneous magnetization of the two dimensional Ising model, Physical Rev. (2) 85 (1952), 808-816.
126. Zenine N., Boukraa S., Hassani S., Maillard J.M., The Fuchsian differential equation of the square Ising model c(3) susceptibility, J. Phys. A: Math. Gen. 37 (2004), 9651-9668, math-ph/0407060.
127. Zenine N., Boukraa S., Hassani S., Maillard J.M., Ising model susceptibility: Fuchsian differential equation for c(4) and its factorization properties, J. Phys. A: Math. Gen. 38 (2005), 4149-4173, cond-mat/0502155.
128. Zenine N., Boukraa S., Hassani S., Maillard J.M., Square lattice Ising model susceptibility: connection matrices and singular behavior of c(3) and c(4), J. Phys. A: Math. Gen. 38 (2005), 9439-9474, hep-th/0506214, math-ph/0506065.
129. Zenine N., Boukraa S., Hassani S., Maillard J.M., Square lattice Ising model susceptibility: series expansion method, and differential equation for c(3), J. Phys. A: Math. Gen. 38 (2005), 1875-1899, hep-ph/0411051.
130. Zhang C., Confluence et phénomènes de Stokes, J. Math. Sci. Univ. Tokyo 3 (1996), 91-107.