Journal of Physics A: Mathematical and Theoretical, 2010
We obtain in exact arithmetic the order 24 linear differential operator L 24 and right hand side ... more We obtain in exact arithmetic the order 24 linear differential operator L 24 and right hand side E (5) of the inhomogeneous equation L 24 (Φ (5) ) = E (5) , where Φ (5) =χ (5) −χ (3) /2 +χ (1) /120 is a linear combination of n-particle contributions to the susceptibility of the square lattice Ising model. In Bostan et al (J. Phys. A: Math. Theor. 42, 275209 (2009)) the operator L 24 (modulo a prime) was shown to factorize into L (left) 12 · L (right) 12
Symmetry, Integrability and Geometry: Methods and Applications, 2007
We recall the form factors f (j) N,N corresponding to the λ-extension C(N, N ; λ) of the two-poin... more We recall the form factors f (j) N,N corresponding to the λ-extension C(N, N ; λ) 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 χ (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 χ (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 + 4w 2 = 0, that occurs in the linear differential equation of χ (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.
We recall the form factors $ f^{(j)}_{N,N}$ corresponding to the $\lambda$-extension $C(N,N; \lam... more We recall the form factors $ f^{(j)}_{N,N}$ corresponding to the $\lambda$-extension $C(N,N; \lambda)$ 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
Journal of Physics A: Mathematical and Theoretical, 2010
We consider the Fuchsian linear differential equation obtained (modulo a prime) forχ (5) , the fi... more We consider the Fuchsian linear differential equation obtained (modulo a prime) forχ (5) , the five-particle contribution to the susceptibility of the square lattice Ising model. We show that one can understand the factorization of the corresponding linear differential operator from calculations using just a single prime. A particular linear combination ofχ (1) andχ (3) can be removed from χ (5) and the resulting series is annihilated by a high order globally nilpotent linear ODE. The corresponding (minimal order) linear differential operator, of order 29, splits into factors of small orders. A fifth-order linear differential operator occurs as the left-most factor of the 'depleted' differential operator and it is shown to be equivalent to the symmetric fourth power of L E , the linear differential operator corresponding to the elliptic integral E. This result generalizes what we have found for the lower order termsχ (3) andχ (4) . We conjecture that a linear differential operator equivalent to a symmetric (n−1) th power of L E occurs as a left-most factor in the minimal order linear differential operators for allχ (n) 's.
Journal of Physics A: Mathematical and Theoretical, 2011
We show that almost all the linear differential operators factors obtained in the analysis of the... more We show that almost all the linear differential operators factors obtained in the analysis of the n-particle contributionsχ (n) 's of the susceptibility of the Ising model, are linear differential operators "associated with elliptic curves".
We use the Hellman-Feynman (HF) and Hypervirial (HV) theorems, to calculate the perturbative coef... more We use the Hellman-Feynman (HF) and Hypervirial (HV) theorems, to calculate the perturbative coefficients of the eigenenergies formal series, in the case of the Coulomb potential with a radial linear term and the radial quartic anharmonic oscillator potential. This calculation method, contrary to the usual Rayleigh-Schrodinger Perturbation Theory (RSPT), does not require the calculation of eigenfunctions coefficients. This method is
Journal of Physics A: Mathematical and Theoretical, 2014
We present a recursive method to generate the expansion of the lattice Green function of the d-di... more We present a recursive method to generate the expansion of the lattice Green function of the d-dimensional face-centred cubic (fcc) lattice. We produce a long series for d = 7. Then we show (and recall) that, in order to obtain the linear differential equation annihilating such a long power series, the most economic way amounts to producing the non-minimal order differential equations. We use the method to obtain the minimal order linear differential equation of the lattice Green function of the seven-dimensional face-centred cubic (fcc) lattice. We give some properties of this irreducible order-eleven differential equation. We show that the differential Galois group of the corresponding operator is included in SO(11, C). This order-eleven operator is non-trivially homomorphic to its adjoint, and we give a "decomposition" of this order-eleven operator in terms of four order-one self-adjoint operators and one order-seven self-adjoint operator. Furthermore, using the Landau conditions on the integral, we forward the regular singularities of the differential equation of the d-dimensional lattice and show that they are all rational numbers. We evaluate the return probability in random walks in the seven-dimensional fcc lattice. We show that the return probability in the d-dimensional fcc lattice decreases as d −2 as the dimension d goes to the infinity.
Differential Geometry and Physics - Proceedings of the 23rd International Conference of Differential Geometric Methods in Theoretical Physics, 2006
We present a simple, but efficient, way to calculate connection matrices between sets of independ... more We present a simple, but efficient, way to calculate connection matrices between sets of independent local solutions, defined at two neighboring singular points, of Fuchsian differential equations of quite large orders, such as those found for the third and fourth contribution (χ (3) and χ (4) ) to the magnetic susceptibility of square lattice Ising model. We use the previous connection matrices to get the exact explicit expressions of all the monodromy matrices of the Fuchsian differential equation for χ (3) (and χ (4) ) expressed in the same basis of solutions. These monodromy matrices are the generators of the differential Galois group of the Fuchsian differential equations for χ (3) (and χ (4) ), whose analysis is just sketched here.
New Trends in Quantum Integrable Systems - Proceedings of the Infinite Analysis 09, 2011
We review developments made since 1959 in the search for a closed form for the susceptibility of ... more We review developments made since 1959 in the search for a closed form for the susceptibility of the Ising model. The expressions for the form factors in terms of the nome q and the modulus k are compared and contrasted. The λ generalized correlations C(M, N ; λ) are defined and explicitly computed in terms of theta functions for M = N = 0, 1.
Journal of Physics A: Mathematical and Theoretical, 2007
We use the recently derived form factor expansions of the diagonal two-point correlation function... more We use the recently derived form factor expansions of the diagonal two-point correlation function of the square Ising model to study the susceptibility for a magnetic field applied only to one diagonal of the lattice, for the isotropic Ising model. We exactly evaluate the one and two particle contributions χ PACS: 02.30.Hq, 02.30.Gp, 02.30.-f, 02.40.Re, 05.50.+q, 05.10.-a, 04.20.Jb AMS Classification scheme numbers: 33E17, 33E05, 33Cxx, 33Dxx, 14Exx, 14Hxx, 34M55, 47E05, 34Lxx, 34Mxx, 14Kxx Keywords: susceptibility of the isotropic square Ising model, two-point correlation functions of the Ising model, singularities of the square Ising model, natural boundary, Fuchsian linear differential equations, complete elliptic integrals.
Journal of Physics A: Mathematical and Theoretical, 2007
We study the Ising model two-point diagonal correlation function C(N, N ) by presenting an expone... more We study the Ising model two-point diagonal correlation function C(N, N ) by presenting an exponential and form factor expansion in an integral representation which differs from the known expansion of Wu, McCoy, Tracy and Barouch. We extend this expansion, weighting, by powers of a variable λ, the j-particle contributions, f (j) N,N . The corresponding λ extension of the twopoint diagonal correlation function, C(N, N ; λ), is shown, for arbitrary λ, to be a solution of the sigma form of the Painlevé VI equation introduced by Jimbo and Miwa. Linear differential equations for the form factors f (j)
Journal of Physics A: Mathematical and Theoretical, 2008
We calculate very long low-and high-temperature series for the susceptibility χ of the square lat... more We calculate very long low-and high-temperature series for the susceptibility χ of the square lattice Ising model as well as very long series for the fiveparticle contribution χ (5) and six-particle contribution χ . These calculations have been made possible by the use of highly optimized polynomial time modular algorithms and a total of more than 150 000 CPU hours on computer clusters. The series for χ (low-and high-temperature regimes), χ (5) and χ are now extended to 2000 terms. In addition, for χ (5) , 10 000 terms of the series are calculated modulo a single prime, and have been used to find the linear ODE satisfied by χ (5) modulo a prime. A diff-Padé analysis of the 2000 terms series for χ and χ (6) confirms to a very high degree of confidence previous conjectures about the location and strength of the singularities of the n-particle components of the susceptibility, up to a small set of 'additional' singularities. The exponents at all the singularities of the Fuchsian linear ODE of χ (5) and the (as yet unknown) ODE of χ (6) are given: they are all rational numbers. We find the presence of singularities at w = 1/2 for the linear ODE of χ (5) , and w 2 = 1/8 for the ODE of χ (6) , which are not singularities of the 'physical' χ (5) and χ , that is to say the series solutions of the ODE's which are analytic at w = 0. Furthermore, analysis of the long series for χ (and χ (6) ) combined with the corresponding long series for the full susceptibility χ yields previously conjectured singularities in some χ (n) , n 7. The exponents at all these singularities are also seen to be rational numbers. We also present a mechanism of resummation of the logarithmic singularities of the χ (n) leading to the known power-law critical behaviour occurring in the full χ , and perform
Journal of Physics A: Mathematical and Theoretical, 2010
We consider the Fuchsian linear differential equation obtained (modulo a prime) forχ (5) , the fi... more We consider the Fuchsian linear differential equation obtained (modulo a prime) forχ (5) , the five-particle contribution to the susceptibility of the square lattice Ising model. We show that one can understand the factorization of the corresponding linear differential operator from calculations using just a single prime. A particular linear combination ofχ (1) andχ (3) can be removed from χ (5) and the resulting series is annihilated by a high order globally nilpotent linear ODE. The corresponding (minimal order) linear differential operator, of order 29, splits into factors of small orders. A fifth-order linear differential operator occurs as the left-most factor of the 'depleted' differential operator and it is shown to be equivalent to the symmetric fourth power of L E , the linear differential operator corresponding to the elliptic integral E. This result generalizes what we have found for the lower order termsχ (3) andχ (4) . We conjecture that a linear differential operator equivalent to a symmetric (n−1) th power of L E occurs as a left-most factor in the minimal order linear differential operators for allχ (n) 's.
Journal of Physics A: Mathematical and Theoretical, 2011
We show that almost all the linear differential operators factors obtained in the analysis of the... more We show that almost all the linear differential operators factors obtained in the analysis of the n-particle contributionsχ (n) 's of the susceptibility of the Ising model, are linear differential operators "associated with elliptic curves".
Journal of Physics A: Mathematical and Theoretical, 2010
We obtain in exact arithmetic the order 24 linear differential operator L 24 and right hand side ... more We obtain in exact arithmetic the order 24 linear differential operator L 24 and right hand side E (5) of the inhomogeneous equation L 24 (Φ (5) ) = E (5) , where Φ (5) =χ (5) −χ (3) /2 +χ (1) /120 is a linear combination of n-particle contributions to the susceptibility of the square lattice Ising model. In Bostan et al (J. Phys. A: Math. Theor. 42, 275209 (2009)) the operator L 24 (modulo a prime) was shown to factorize into L (left) 12 · L (right) 12
Journal of Physics A: Mathematical and Theoretical, 2009
We recall various multiple integrals with one parameter, related to the isotropic square Ising mo... more We recall various multiple integrals with one parameter, related to the isotropic square Ising model, and corresponding, respectively, to the n-particle contributions of the magnetic susceptibility, to the (lattice) form factors, to the two-point correlation functions and to their λ-extensions. The univariate analytic functions defined by these integrals are holonomic and even G-functions: they satisfy Fuchsian linear differential equations with polynomial coefficients and have some arithmetic properties. We recall the explicit forms, found in previous work, of these Fuchsian equations, as well as their russian-doll and direct sum structures. These differential operators are very selected Fuchsian linear differential operators, and their remarkable properties have a deep geometrical origin: they are all globally nilpotent, or, sometimes, even have zero p-curvature. We also display miscellaneous examples of globally nilpotent operators emerging from enumerative combinatorics problems for which no integral representation is yet known. Focusing on the factorised parts of all these operators, we find out that the global nilpotence of the factors (resp. p-curvature nullity) corresponds to a set of selected structures of algebraic geometry: elliptic curves, modular curves, curves of genus five, six, . . . , and even a remarkable weight-1 modular form emerging in the three-particle contribution χ (3) of the magnetic susceptibility of the square Ising model. Noticeably, this associated weight-1 modular form is also seen in the factors of the differential operator for another n-fold integral of the Ising class, Φ
Journal of Physics A: Mathematical and Theoretical, 2007
We briefly ‡, recall the Fuchs-Painlevé elliptic representation of Painlevé VI. We then show that... more We briefly ‡, recall the Fuchs-Painlevé elliptic representation of Painlevé VI. We then show that the polynomiality of the expressions of the correlation functions (and form factors) in terms of the complete elliptic integral of the first and second kind, K and E, is a straight consequence of the fact that the differential operators corresponding to the entries of Toeplitz-like determinants, are equivalent to the second order operator L E which has E as solution (or, for off-diagonal correlations to the direct sum of L E and d/dt). We show that this can be generalized, mutatis mutandis, to the anisotropic Ising model. The singled-out second order linear differential operator L E is replaced by an isomonodromic system of two third-order linear partial differential operators associated with Π 1 , the Jacobi's form of the complete elliptic integral of the third kind (or equivalently two second order linear partial differential operators associated with Appell functions, where one of these operators can be seen as a deformation of L E ). We finally explore the generalizations, to the anisotropic Ising models, of the links we made, in two previous papers, between Painlevé non-linear ODE's, Fuchsian linear ODE's and elliptic curves. In particular the elliptic representation of Painlevé VI has to be generalized to an "Appellian" representation of Garnier systems.
Journal of Physics A: Mathematical and Theoretical, 2010
We obtain in exact arithmetic the order 24 linear differential operator L 24 and right hand side ... more We obtain in exact arithmetic the order 24 linear differential operator L 24 and right hand side E (5) of the inhomogeneous equation L 24 (Φ (5) ) = E (5) , where Φ (5) =χ (5) −χ (3) /2 +χ (1) /120 is a linear combination of n-particle contributions to the susceptibility of the square lattice Ising model. In Bostan et al (J. Phys. A: Math. Theor. 42, 275209 (2009)) the operator L 24 (modulo a prime) was shown to factorize into L (left) 12 · L (right) 12
Symmetry, Integrability and Geometry: Methods and Applications, 2007
We recall the form factors f (j) N,N corresponding to the λ-extension C(N, N ; λ) of the two-poin... more We recall the form factors f (j) N,N corresponding to the λ-extension C(N, N ; λ) 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 χ (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 χ (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 + 4w 2 = 0, that occurs in the linear differential equation of χ (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.
We recall the form factors $ f^{(j)}_{N,N}$ corresponding to the $\lambda$-extension $C(N,N; \lam... more We recall the form factors $ f^{(j)}_{N,N}$ corresponding to the $\lambda$-extension $C(N,N; \lambda)$ 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
Journal of Physics A: Mathematical and Theoretical, 2010
We consider the Fuchsian linear differential equation obtained (modulo a prime) forχ (5) , the fi... more We consider the Fuchsian linear differential equation obtained (modulo a prime) forχ (5) , the five-particle contribution to the susceptibility of the square lattice Ising model. We show that one can understand the factorization of the corresponding linear differential operator from calculations using just a single prime. A particular linear combination ofχ (1) andχ (3) can be removed from χ (5) and the resulting series is annihilated by a high order globally nilpotent linear ODE. The corresponding (minimal order) linear differential operator, of order 29, splits into factors of small orders. A fifth-order linear differential operator occurs as the left-most factor of the 'depleted' differential operator and it is shown to be equivalent to the symmetric fourth power of L E , the linear differential operator corresponding to the elliptic integral E. This result generalizes what we have found for the lower order termsχ (3) andχ (4) . We conjecture that a linear differential operator equivalent to a symmetric (n−1) th power of L E occurs as a left-most factor in the minimal order linear differential operators for allχ (n) 's.
Journal of Physics A: Mathematical and Theoretical, 2011
We show that almost all the linear differential operators factors obtained in the analysis of the... more We show that almost all the linear differential operators factors obtained in the analysis of the n-particle contributionsχ (n) 's of the susceptibility of the Ising model, are linear differential operators "associated with elliptic curves".
We use the Hellman-Feynman (HF) and Hypervirial (HV) theorems, to calculate the perturbative coef... more We use the Hellman-Feynman (HF) and Hypervirial (HV) theorems, to calculate the perturbative coefficients of the eigenenergies formal series, in the case of the Coulomb potential with a radial linear term and the radial quartic anharmonic oscillator potential. This calculation method, contrary to the usual Rayleigh-Schrodinger Perturbation Theory (RSPT), does not require the calculation of eigenfunctions coefficients. This method is
Journal of Physics A: Mathematical and Theoretical, 2014
We present a recursive method to generate the expansion of the lattice Green function of the d-di... more We present a recursive method to generate the expansion of the lattice Green function of the d-dimensional face-centred cubic (fcc) lattice. We produce a long series for d = 7. Then we show (and recall) that, in order to obtain the linear differential equation annihilating such a long power series, the most economic way amounts to producing the non-minimal order differential equations. We use the method to obtain the minimal order linear differential equation of the lattice Green function of the seven-dimensional face-centred cubic (fcc) lattice. We give some properties of this irreducible order-eleven differential equation. We show that the differential Galois group of the corresponding operator is included in SO(11, C). This order-eleven operator is non-trivially homomorphic to its adjoint, and we give a "decomposition" of this order-eleven operator in terms of four order-one self-adjoint operators and one order-seven self-adjoint operator. Furthermore, using the Landau conditions on the integral, we forward the regular singularities of the differential equation of the d-dimensional lattice and show that they are all rational numbers. We evaluate the return probability in random walks in the seven-dimensional fcc lattice. We show that the return probability in the d-dimensional fcc lattice decreases as d −2 as the dimension d goes to the infinity.
Differential Geometry and Physics - Proceedings of the 23rd International Conference of Differential Geometric Methods in Theoretical Physics, 2006
We present a simple, but efficient, way to calculate connection matrices between sets of independ... more We present a simple, but efficient, way to calculate connection matrices between sets of independent local solutions, defined at two neighboring singular points, of Fuchsian differential equations of quite large orders, such as those found for the third and fourth contribution (χ (3) and χ (4) ) to the magnetic susceptibility of square lattice Ising model. We use the previous connection matrices to get the exact explicit expressions of all the monodromy matrices of the Fuchsian differential equation for χ (3) (and χ (4) ) expressed in the same basis of solutions. These monodromy matrices are the generators of the differential Galois group of the Fuchsian differential equations for χ (3) (and χ (4) ), whose analysis is just sketched here.
New Trends in Quantum Integrable Systems - Proceedings of the Infinite Analysis 09, 2011
We review developments made since 1959 in the search for a closed form for the susceptibility of ... more We review developments made since 1959 in the search for a closed form for the susceptibility of the Ising model. The expressions for the form factors in terms of the nome q and the modulus k are compared and contrasted. The λ generalized correlations C(M, N ; λ) are defined and explicitly computed in terms of theta functions for M = N = 0, 1.
Journal of Physics A: Mathematical and Theoretical, 2007
We use the recently derived form factor expansions of the diagonal two-point correlation function... more We use the recently derived form factor expansions of the diagonal two-point correlation function of the square Ising model to study the susceptibility for a magnetic field applied only to one diagonal of the lattice, for the isotropic Ising model. We exactly evaluate the one and two particle contributions χ PACS: 02.30.Hq, 02.30.Gp, 02.30.-f, 02.40.Re, 05.50.+q, 05.10.-a, 04.20.Jb AMS Classification scheme numbers: 33E17, 33E05, 33Cxx, 33Dxx, 14Exx, 14Hxx, 34M55, 47E05, 34Lxx, 34Mxx, 14Kxx Keywords: susceptibility of the isotropic square Ising model, two-point correlation functions of the Ising model, singularities of the square Ising model, natural boundary, Fuchsian linear differential equations, complete elliptic integrals.
Journal of Physics A: Mathematical and Theoretical, 2007
We study the Ising model two-point diagonal correlation function C(N, N ) by presenting an expone... more We study the Ising model two-point diagonal correlation function C(N, N ) by presenting an exponential and form factor expansion in an integral representation which differs from the known expansion of Wu, McCoy, Tracy and Barouch. We extend this expansion, weighting, by powers of a variable λ, the j-particle contributions, f (j) N,N . The corresponding λ extension of the twopoint diagonal correlation function, C(N, N ; λ), is shown, for arbitrary λ, to be a solution of the sigma form of the Painlevé VI equation introduced by Jimbo and Miwa. Linear differential equations for the form factors f (j)
Journal of Physics A: Mathematical and Theoretical, 2008
We calculate very long low-and high-temperature series for the susceptibility χ of the square lat... more We calculate very long low-and high-temperature series for the susceptibility χ of the square lattice Ising model as well as very long series for the fiveparticle contribution χ (5) and six-particle contribution χ . These calculations have been made possible by the use of highly optimized polynomial time modular algorithms and a total of more than 150 000 CPU hours on computer clusters. The series for χ (low-and high-temperature regimes), χ (5) and χ are now extended to 2000 terms. In addition, for χ (5) , 10 000 terms of the series are calculated modulo a single prime, and have been used to find the linear ODE satisfied by χ (5) modulo a prime. A diff-Padé analysis of the 2000 terms series for χ and χ (6) confirms to a very high degree of confidence previous conjectures about the location and strength of the singularities of the n-particle components of the susceptibility, up to a small set of 'additional' singularities. The exponents at all the singularities of the Fuchsian linear ODE of χ (5) and the (as yet unknown) ODE of χ (6) are given: they are all rational numbers. We find the presence of singularities at w = 1/2 for the linear ODE of χ (5) , and w 2 = 1/8 for the ODE of χ (6) , which are not singularities of the 'physical' χ (5) and χ , that is to say the series solutions of the ODE's which are analytic at w = 0. Furthermore, analysis of the long series for χ (and χ (6) ) combined with the corresponding long series for the full susceptibility χ yields previously conjectured singularities in some χ (n) , n 7. The exponents at all these singularities are also seen to be rational numbers. We also present a mechanism of resummation of the logarithmic singularities of the χ (n) leading to the known power-law critical behaviour occurring in the full χ , and perform
Journal of Physics A: Mathematical and Theoretical, 2010
We consider the Fuchsian linear differential equation obtained (modulo a prime) forχ (5) , the fi... more We consider the Fuchsian linear differential equation obtained (modulo a prime) forχ (5) , the five-particle contribution to the susceptibility of the square lattice Ising model. We show that one can understand the factorization of the corresponding linear differential operator from calculations using just a single prime. A particular linear combination ofχ (1) andχ (3) can be removed from χ (5) and the resulting series is annihilated by a high order globally nilpotent linear ODE. The corresponding (minimal order) linear differential operator, of order 29, splits into factors of small orders. A fifth-order linear differential operator occurs as the left-most factor of the 'depleted' differential operator and it is shown to be equivalent to the symmetric fourth power of L E , the linear differential operator corresponding to the elliptic integral E. This result generalizes what we have found for the lower order termsχ (3) andχ (4) . We conjecture that a linear differential operator equivalent to a symmetric (n−1) th power of L E occurs as a left-most factor in the minimal order linear differential operators for allχ (n) 's.
Journal of Physics A: Mathematical and Theoretical, 2011
We show that almost all the linear differential operators factors obtained in the analysis of the... more We show that almost all the linear differential operators factors obtained in the analysis of the n-particle contributionsχ (n) 's of the susceptibility of the Ising model, are linear differential operators "associated with elliptic curves".
Journal of Physics A: Mathematical and Theoretical, 2010
We obtain in exact arithmetic the order 24 linear differential operator L 24 and right hand side ... more We obtain in exact arithmetic the order 24 linear differential operator L 24 and right hand side E (5) of the inhomogeneous equation L 24 (Φ (5) ) = E (5) , where Φ (5) =χ (5) −χ (3) /2 +χ (1) /120 is a linear combination of n-particle contributions to the susceptibility of the square lattice Ising model. In Bostan et al (J. Phys. A: Math. Theor. 42, 275209 (2009)) the operator L 24 (modulo a prime) was shown to factorize into L (left) 12 · L (right) 12
Journal of Physics A: Mathematical and Theoretical, 2009
We recall various multiple integrals with one parameter, related to the isotropic square Ising mo... more We recall various multiple integrals with one parameter, related to the isotropic square Ising model, and corresponding, respectively, to the n-particle contributions of the magnetic susceptibility, to the (lattice) form factors, to the two-point correlation functions and to their λ-extensions. The univariate analytic functions defined by these integrals are holonomic and even G-functions: they satisfy Fuchsian linear differential equations with polynomial coefficients and have some arithmetic properties. We recall the explicit forms, found in previous work, of these Fuchsian equations, as well as their russian-doll and direct sum structures. These differential operators are very selected Fuchsian linear differential operators, and their remarkable properties have a deep geometrical origin: they are all globally nilpotent, or, sometimes, even have zero p-curvature. We also display miscellaneous examples of globally nilpotent operators emerging from enumerative combinatorics problems for which no integral representation is yet known. Focusing on the factorised parts of all these operators, we find out that the global nilpotence of the factors (resp. p-curvature nullity) corresponds to a set of selected structures of algebraic geometry: elliptic curves, modular curves, curves of genus five, six, . . . , and even a remarkable weight-1 modular form emerging in the three-particle contribution χ (3) of the magnetic susceptibility of the square Ising model. Noticeably, this associated weight-1 modular form is also seen in the factors of the differential operator for another n-fold integral of the Ising class, Φ
Journal of Physics A: Mathematical and Theoretical, 2007
We briefly ‡, recall the Fuchs-Painlevé elliptic representation of Painlevé VI. We then show that... more We briefly ‡, recall the Fuchs-Painlevé elliptic representation of Painlevé VI. We then show that the polynomiality of the expressions of the correlation functions (and form factors) in terms of the complete elliptic integral of the first and second kind, K and E, is a straight consequence of the fact that the differential operators corresponding to the entries of Toeplitz-like determinants, are equivalent to the second order operator L E which has E as solution (or, for off-diagonal correlations to the direct sum of L E and d/dt). We show that this can be generalized, mutatis mutandis, to the anisotropic Ising model. The singled-out second order linear differential operator L E is replaced by an isomonodromic system of two third-order linear partial differential operators associated with Π 1 , the Jacobi's form of the complete elliptic integral of the third kind (or equivalently two second order linear partial differential operators associated with Appell functions, where one of these operators can be seen as a deformation of L E ). We finally explore the generalizations, to the anisotropic Ising models, of the links we made, in two previous papers, between Painlevé non-linear ODE's, Fuchsian linear ODE's and elliptic curves. In particular the elliptic representation of Painlevé VI has to be generalized to an "Appellian" representation of Garnier systems.
Uploads
Papers by N. Zenine