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J. -M. Maillard

Publications and source records attributed to J. -M. Maillard.

At least 19 recordsLinked to original sources
arXiv2022

Symmetries of non-linear ODEs: lambda extensions of the Ising correlations

This paper provides several illustrations of the numerous remarkable properties of the lambda-extensions of the two-point correlation functions of the Ising model, sheding some light on the non-linear ODEs of the Painlevé type. We first show that this concept also exists for the factors of the two-point correlation functions focusing, for pedagogical reasons, on two examples namely C(0,5) and C(2,5) at $ν= -k$. We then display, in a learn-by-example approach, some of the puzzling properties and structures of these lambda-extensions: for an infinite set of (algebraic) values of $ λ$ these power series become algebraic functions, and for a finite set of (rational) values of lambda they become D-finite functions, more precisely polynomials (of different degrees) in the complete elliptic integrals of the first and second kind K and E. For generic values of $ λ$ these power series are not D-finite, they are differentially algebraic. For an infinite number of other (rational) values of $ λ$ these power series are globally bounded series, thus providing an example of an infinite number of globally bounded differentially algebraic series. Finally, taking the example of a product of two diagonal two-point correlation functions, we suggest that many more families of non-linear ODEs of the Painlevé type remain to be discovered on the two-dimensional Ising model, as well as their structures, and in particular their associated lambda extensions. The question of their possible reduction, after complicated transformations, to Okamoto sigma forms of Painlevé VI remains an extremely difficult challenge.

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arXiv2022

Factorization of Ising correlations C(M,N) for $ ν= \, -k$ and M+N odd, $M \le N$, $T < T_c$ and their lambda extensions

We study the factorizations of Ising low-temperature correlations C(M,N) for $ν=-k$ and M+N odd, $M \le N$, for both the cases $M\neq 0$ where there are two factors, and $M=0$ where there are four factors. We find that the two factors for $ M \neq 0$ satisfy the same non-linear differential equation and, similarly, for M=0 the four factors each satisfy Okamoto sigma-form of Painlevé VI equations with the same Okamoto parameters. Using a Landen transformation we show, for $M\neq 0$, that the previous non-linear differential equation can actually be reduced to an Okamoto sigma-form of Painlevé VI equation. For both the two and four factor case, we find that there is a one parameter family of boundary conditions on the Okamoto sigma-form of Painlevé VI equations which generalizes the factorization of the correlations C(M,N) to an additive decomposition of the corresponding sigma's solutions of the Okamoto sigma-form of Painlevé VI equation which we call lambda extensions. At a special value of the parameter, the lambda-extensions of the factors of C(M,N) reduce to homogeneous polynomials in the complete elliptic functions of the first and second kind. We also generalize some Tracy-Widom (Painlevé V) relations between the sum and difference of sigma's to this Painlevé VI framework.

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arXiv2017

Schwarzian conditions for linear differential operators with selected differential Galois groups (unabridged version)

We show that non-linear Schwarzian differential equations emerging from covariance symmetry conditions imposed on linear differential operators with hypergeometric function solutions, can be generalized to arbitrary order linear differential operators with polynomial coefficients having selected differential Galois groups. For order three and order four linear differential operators we show that this pullback invariance up to conjugation eventually reduces to symmetric powers of an underlying order-two operator. We give, precisely, the conditions to have modular correspondences solutions for such Schwarzian differential equations, which was an open question in a previous paper. We analyze in detail a pullbacked hypergeometric example generalizing modular forms, that ushers a pullback invariance up to operator homomorphisms. We expect this new concept to be well-suited in physics and enumerative combinatorics. We finally consider the more general problem of the equivalence of two different order-four linear differential Calabi-Yau operators up to pullbacks and conjugation, and clarify the cases where they have the same Yukawa couplings.

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arXiv2017

Modular forms, Schwarzian conditions, and symmetries of differential equations in physics

We give examples of infinite order rational transformations that leave linear differential equations covariant. These examples are non-trivial yet simple enough illustrations of exact representations of the renormalization group. We first illustrate covariance properties on order-two linear differential operators associated with identities relating the same $_2F_1$ hypergeometric function with different rational pullbacks. We provide two new and more general results of the previous covariance by rational functions: a new Heun function example and a higher genus $_2F_1$ hypergeometric function example. We then focus on identities relating the same hypergeometric function with two different algebraic pullback transformations: such remarkable identities correspond to modular forms, the algebraic transformations being solution of another differentially algebraic Schwarzian equation that emerged in a paper by Casale. Further, we show that the first differentially algebraic equation can be seen as a subcase of the last Schwarzian differential condition, the restriction corresponding to a factorization condition of some associated order-two linear differential operator. Finally, we also explore generalizations of these results, for instance, to $_3F_2$, hypergeometric functions, and show that one just reduces to the previous $_2F_1$ cases through a Clausen identity. In a $_2F_1$ hypergeometric framework the Schwarzian condition encapsulates all the modular forms and modular equations of the theory of elliptic curves, but these two conditions are actually richer than elliptic curves or $_2F_1$ hypergeometric functions, as can be seen on the Heun and higher genus example. This work is a strong incentive to develop more differentially algebraic symmetry analysis in physics.

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arXiv2013

Ising n-fold integrals as diagonals of rational functions and integrality of series expansions

We show that the n-fold integrals $χ^{(n)}$ of the magnetic susceptibility of the Ising model, as well as various other n-fold integrals of the "Ising class", or n-fold integrals from enumerative combinatorics, like lattice Green functions, correspond to a distinguished class of function generalising algebraic functions: they are actually diagonals of rational functions. As a consequence, the power series expansions of the, analytic at x=0, solutions of these linear differential equations "Derived From Geometry" are globally bounded, which means that, after just one rescaling of the expansion variable, they can be cast into series expansions with integer coefficients. We also give several results showing that the unique analytical solution of Calabi-Yau ODEs, and, more generally, Picard-Fuchs linear ODEs, with solutions of maximal weights, are always diagonal of rational functions. Besides, in a more enumerative combinatorics context, generating functions whose coefficients are expressed in terms of nested sums of products of binomial terms can also be shown to be diagonals of rational functions. We finally address the question of the relations between the notion of integrality (series with integer coefficients, or, more generally, globally bounded series) and the modularity of ODEs.

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arXiv2012

Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity

We show that the n-fold integrals $χ^{(n)}$ of the magnetic susceptibility of the Ising model, as well as various other n-fold integrals of the "Ising class", or n-fold integrals from enumerative combinatorics, like lattice Green functions, are actually diagonals of rational functions. As a consequence, the power series expansions of these solutions of linear differential equations "Derived From Geometry" are globally bounded, which means that, after just one rescaling of the expansion variable, they can be cast into series expansions with integer coefficients. Besides, in a more enumerative combinatorics context, we show that generating functions whose coefficients are expressed in terms of nested sums of products of binomial terms can also be shown to be diagonals of rational functions. We give a large set of results illustrating the fact that the unique analytical solution of Calabi-Yau ODEs, and more generally of MUM ODEs, is, almost always, diagonal of rational functions. We revisit Christol's conjecture that globally bounded series of G-operators are necessarily diagonals of rational functions. We provide a large set of examples of globally bounded series, or series with integer coefficients, associated with modular forms, or Hadamard product of modular forms, or associated with Calabi-Yau ODEs, underlying the concept of modularity. We finally address the question of the relations between the notion of integrality (series with integer coefficients, or, more generally, globally bounded series) and the modularity (in particular integrality of the Taylor coefficients of mirror map), introducing new representations of Yukawa couplings.

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arXiv2010

The Ising model: from elliptic curves to modular forms and Calabi-Yau equations

We show that almost all the linear differential operators factors obtained in the analysis of the n-particle contribution of the susceptibility of the Ising model for $\, n \le 6$, are operators "associated with elliptic curves". Beyond the simplest factors which are homomorphic to symmetric powers of the second order operator associated with the complete elliptic integral E, the second and third order differential operators can actually be interpreted as modular forms of the elliptic curve of the Ising model. A last order-four globally nilpotent operator is not reducible to this elliptic curve, modular forms scheme. It is shown to actually correspond to a natural generalization of this elliptic curve, modular forms scheme, with the emergence of a Calabi-Yau equation, corresponding to a selected $_4F_3$ hypergeometric function which can also be seen as a Hadamard product of the complete elliptic integral K, with a remarkably simple algebraic pull-back, the corresponding Calabi-Yau fourth-order differential operator having a symplectic differential Galois group SP(4,C). The associated mirror maps and higher order Schwarzian ODEs has an exact (isogenies) representation of the generators of the renormalization group, extending the modular group SL(2,Z) to a GL(2, Z) symmetry group.

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arXiv2010

Square lattice Ising model $\tildeχ^{(5)}$ ODE in exact arithmetic

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 $Φ^{(5)} =\tildeχ^{(5)}-\tildeχ^{(3)}/2+\tildeχ^{(1)}/120$ 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. {\bf 42}, 275209 (2009)) the operator $L_{24}$ (modulo a prime) was shown to factorize into $L_{12}^{(\rm left)} \cdot L_{12}^{(\rm right)}$; here we prove that no further factorization of the order 12 operator $L_{12}^{(\rm left)}$ is possible. We use the exact ODE to obtain the behaviour of $\tildeχ^{(5)}$ at the ferromagnetic critical point and to obtain a limited number of analytic continuations of $\tildeχ^{(5)}$ beyond the principal disk defined by its high temperature series. Contrary to a speculation in Boukraa, et al (J. Phys. A: Math. Theor. {\bf 41} 455202 (2008)), we find that $\tildeχ^{(5)}$ is singular at $w=1/2$ on an infinite number of branches.

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arXiv2009

High order Fuchsian equations for the square lattice Ising model: $χ^{(6)}$

This paper deals with $\tildeχ^{(6)}$, the six-particle contribution to the magnetic susceptibility of the square lattice Ising model. We have generated, modulo a prime, series coefficients for $\tildeχ^{(6)}$. The length of the series is sufficient to produce the corresponding Fuchsian linear differential equation (modulo a prime). We obtain the Fuchsian linear differential equation that annihilates the "depleted" series $Φ^{(6)}=\tildeχ^{(6)} - {2 \over 3} \tildeχ^{(4)} + {2 \over 45} \tildeχ^{(2)}$. The factorization of the corresponding differential operator is performed using a method of factorization modulo a prime introduced in a previous paper. The "depleted" differential operator is shown to have a structure similar to the corresponding operator for $\tildeχ^{(5)}$. It splits into factors of smaller orders, with the left-most factor of order six being equivalent to the symmetric fifth power of the linear differential operator corresponding to the elliptic integral $E$. The right-most factor has a direct sum structure, and using series calculated modulo several primes, all the factors in the direct sum have been reconstructed in exact arithmetics.

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arXiv2009

A birational mapping with a strange attractor: Post critical set and covariant curves

We consider some two-dimensional birational transformations. One of them is a birational deformation of the Hénon map. For some of these birational mappings, the post critical set (i.e. the iterates of the critical set) is infinite and we show that this gives straightforwardly the algebraic covariant curves of the transformation when they exist. These covariant curves are used to build the preserved meromorphic two-form. One may have also an infinite post critical set yielding a covariant curve which is not algebraic (transcendent). For two of the birational mappings considered, the post critical set is not infinite and we claim that there is no algebraic covariant curve and no preserved meromorphic two-form. For these two mappings with non infinite post critical sets, attracting sets occur and we show that they pass the usual tests (Lyapunov exponents and the fractal dimension) for being strange attractors. The strange attractor of one of these two mappings is unbounded.

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arXiv2009

High order Fuchsian equations for the square lattice Ising model: $\tildeχ^{(5)}$

We consider the Fuchsian linear differential equation obtained (modulo a prime) for $\tildeχ^{(5)}$, 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 of $\tildeχ^{(1)}$ and $\tildeχ^{(3)}$ can be removed from $\tildeχ^{(5)}$ 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 terms $\tildeχ^{(3)}$ and $\tildeχ^{(4)}$. 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 all $\tildeχ^{(n)}$'s.

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arXiv2008

Globally nilpotent differential operators and the square Ising model

We recall various multiple integrals 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 lambda-extensions. 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. 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. Focusing on the factorised parts of all these operators, we find out that the global nilpotence of the factors corresponds to a set of selected structures of algebraic geometry: elliptic curves, modular curves, 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. In the case where we do not have G-functions, but Hamburger functions (one irregular singularity at 0 or $ \infty$) that correspond to the confluence of singularities in the scaling limit, the p-curvature is also found to verify new structures associated with simple deformations of the nilpotent property.

math-ph↗
arXiv2008

Birational Mappings and Matrix Sub-algebra from the Chiral Potts Model

We study birational transformations of the projective space originating from lattice statistical mechanics, specifically from various chiral Potts models. Associating these models to \emph{stable patterns} and \emph{signed-patterns}, we give general results which allow us to find \emph{all} chiral $q$-state spin-edge Potts models when the number of states $q$ is a prime or the square of a prime, as well as several $q$-dependent family of models. We also prove the absence of monocolor stable signed-pattern with more than four states. This demonstrates a conjecture about cyclic Hadamard matrices in a particular case. The birational transformations associated to these lattice spin-edge models show complexity reduction. In particular we recover a one-parameter family of integrable transformations, for which we give a matrix representation

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arXiv2008

Experimental mathematics on the magnetic susceptibility of the square lattice Ising model

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 five-particle contribution $χ^{(5)}$ and six-particle contribution $χ^{(6)}$. These calculations have been made possible by the use of highly optimized polynomial time modular algorithms and a total of more than 150000 CPU hours on computer clusters. For $χ^{(5)}$ 10000 terms of the series are calculated {\it modulo} a single prime, and have been used to find the linear ODE satisfied by $χ^{(5)}$ {\it modulo} a prime. A diff-Padé analysis of 2000 terms series for $χ^{(5)}$ 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. 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 {\it not} singularities of the ``physical'' $χ^{(5)}$ and $χ^{(6)},$ that is to say the series-solutions of the ODE's which are analytic at $w =0$. Furthermore, analysis of the long series for $χ^{(5)}$ (and $χ^{(6)}$) combined with the corresponding long series for the full susceptibility $χ$ yields previously conjectured singularities in some $χ^{(n)}$, $n \ge 7$. 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 a power spectrum analysis giving strong arguments in favor of the existence of a natural boundary for the full susceptibility $χ$.

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arXiv2007

Singularities of $n$-fold integrals of the Ising class and the theory of elliptic curves

We introduce some multiple integrals that are 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 equation satisfied by these multiple integrals for $ n=1, 2, 3, 4$ and only modulo some primes for $ n=5$ and $ 6$, thus providing a large set of (possible) new singularities of the $χ^{(n)}$. We discuss the singularity structure for these multiple integrals by solving the Landau conditions. We find that the singularities of the associated ODEs identify (up to $n= 6$) with the leading pinch Landau singularities. The second remarkable obtained feature is that the singularities of the ODEs associated with the multiple integrals reduce to the singularities of the ODEs associated with a {\em finite number of one dimensional integrals}. Among the singularities found, we underline the fact that the quadratic polynomial condition $ 1+3 w +4 w^2 = 0$, that occurs in the linear differential equation of $ χ^{(3)}$, actually corresponds to a remarkable property of selected elliptic curves, namely the occurrence of complex multiplication. The interpretation of complex multiplication for elliptic curves as complex fixed points of the selected generators of the renormalization group, namely isogenies of elliptic curves, is sketched. Most of the other singularities occurring in our multiple integrals are not related to complex multiplication situations, suggesting an interpretation in terms of (motivic) mathematical structures beyond the theory of elliptic curves.

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arXiv2007

The diagonal Ising susceptibility

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 $χ_{d}^{(1)}$ and $χ_{d}^{(2)}$ of the corresponding susceptibility, and obtain linear differential equations for the three and four particle contributions, as well as the five particle contribution $χ^{(5)}_d(t)$, but only modulo a given prime. We use these exact linear differential equations to show that, not only the russian-doll structure, but also the direct sum structure on the linear differential operators for the $ n$-particle contributions $χ_{d}^{(n)}$ are quite directly inherited from the direct sum structure on the form factors $ f^{(n)}$. We show that the $ n^{th}$ particle contributions $χ_{d}^{(n)}$ have their singularities at roots of unity. These singularities become dense on the unit circle $|\sinh2E_v/kT \sinh 2E_h/kT|=1$ as $ n\to \infty$.

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arXiv2007

Fuchs versus Painlevé

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$ being 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.

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