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J. -M. Maillard

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

34 records · Page 2Linked to original sources
arXiv2007

Landau singularities and singularities of holonomic integrals of the Ising class

We consider families of multiple and simple integrals of the ``Ising class'' and the linear ordinary differential equations with polynomial coefficients they are solutions of. We compare the full set of singularities given by the roots of the head polynomial of these linear ODE's and the subset of singularities occurring in the integrals, with the singularities obtained from the Landau conditions. For these Ising class integrals, we show that the Landau conditions can be worked out, either to give the singularities of the corresponding linear differential equation or the singularities occurring in the integral. The singular behavior of these integrals is obtained in the self-dual variable $w= s/2/(1+s^2)$, with $s= \sinh(2K)$, where $K=J/kT$ is the usual Ising model coupling constant. Switching to the variable $s$, we show that the singularities of the analytic continuation of series expansions of these integrals actually break the Kramers-Wannier duality. We revisit the singular behavior (J. Phys. A {\bf 38} (2005) 9439-9474) of the third contribution to the magnetic susceptibility of Ising model $χ^{(3)}$ at the points $1+3w+4w^2= 0$ and show that $χ^{(3)}(s)$ is not singular at the corresponding points inside the unit circle $| s |=1$, while its analytical continuation in the variable $s$ is actually singular at the corresponding points $ 2+s+s^2=0$ oustside the unit circle ($| s | > 1$).

math-ph↗
arXiv2006

Holonomy of the Ising model form factors

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 two-point diagonal correlation function, $ C(N,N; λ)$, is shown, for arbitrary $λ$, to be a solution of the sigma form of the Painlev{é} VI equation introduced by Jimbo and Miwa. Linear differential equations for the form factors $ f^{(j)}_{N,N}$ are obtained and shown to have both a ``Russian doll'' nesting, and a decomposition of the differential operators as a direct sum of operators equivalent to symmetric powers of the differential operator of the elliptic integral $ E$. Each $ f^{(j)}_{N,N}$ is expressed polynomially in terms of the elliptic integrals $ E$ and $ K$. The scaling limit of these differential operators breaks the direct sum structure but not the ``Russian doll'' structure. The previous $ λ$-extensions, $ C(N,N; λ)$ are, for singled-out values $ λ= \cos(πm/n)$ ($m, n$ integers), also solutions of linear differential equations. These solutions of Painlevé VI are actually algebraic functions, being associated with modular curves.

math-ph↗
arXiv2006

Painleve versus Fuchs

The sigma form of the Painlev{é} VI equation contains four arbitrary parameters and generically the solutions can be said to be genuinely ``nonlinear'' because they do not satisfy linear differential equations of finite order. However, when there are certain restrictions on the four parameters there exist one parameter families of solutions which do satisfy (Fuchsian) differential equations of finite order. We here study this phenomena of Fuchsian solutions to the Painlev{é} equation with a focus on the particular PVI equation which is satisfied by the diagonal correlation function C(N,N) of the Ising model. We obtain Fuchsian equations of order $N+1$ for C(N,N) and show that the equation for C(N,N) is equivalent to the $N^{th}$ symmetric power of the equation for the elliptic integral $E$. We show that these Fuchsian equations correspond to rational algebraic curves with an additional Riccati structure and we show that the Malmquist Hamiltonian $p,q$ variables are rational functions in complete elliptic integrals. Fuchsian equations for off diagonal correlations $C(N,M)$ are given which extend our considerations to discrete generalizations of Painlev{é}.

math-ph↗
arXiv2005

Differential Galois groups of high order Fuchsian ODE's

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.

math-ph↗
arXiv2005

Beyond series expansions: mathematical structures for the susceptibility of the square lattice Ising model

We first study the properties of the Fuchsian ordinary differential equations for the three and four-particle contributions $ χ^{(3)}$ and $ χ^{(4)}$ of the square lattice Ising model susceptibility. An analysis of some mathematical properties of these Fuchsian differential equations is sketched. For instance, we study the factorization properties of the corresponding linear differential operators, and consider the singularities of the three and four-particle contributions $ χ^{(3)}$ and $ χ^{(4)}$, versus the singularities of the associated Fuchsian ordinary differential equations, which actually exhibit new ``Landau-like'' singularities. We sketch the analysis of the corresponding differential Galois groups. In particular we provide a simple, but efficient, method to calculate the so-called ``connection matrices'' (between two neighboring singularities) and deduce the singular behaviors of $ χ^{(3)}$ and $ χ^{(4)}$. We provide a set of comments and speculations on the Fuchsian ordinary differential equations associated with the $ n$-particle contributions $ χ^{(n)}$ and address the problem of the apparent discrepancy between such a holonomic approach and some scaling results deduced from a Painlevé oriented approach.

hep-th↗
arXiv2005

Square lattice Ising model susceptibility: connection matrices and singular behavior of $χ^{(3)}$ and $χ^{(4)}$

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 deduce all the critical behaviors of the solutions $χ^{(3)}$ and $χ^{(4)}$, as well as the asymptotic behavior of the coefficients in the corresponding series expansions. We confirm that the newly found quadratic number singularities of the Fuchsian ODE associated to $χ^{(3)}$ are not singularities of the particular solution $χ^{(3)}$ itself. We use the previous connection matrices to get the exact 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. As far as the physics implications of the solutions are concerned, we find challenging qualitative differences when comparing the corrections to scaling for the full susceptibillity $χ$ at high temperature (resp. low temperature) and the first two terms $χ^{(1)}$ and $χ^{(3)}$ (resp. $χ^{(2)}$ and $χ^{(4)}$) .

math-ph↗
arXiv2005

Post-critical set and non existence of preserved meromorphic two-forms

We present a family of birational transformations in $ CP_2$ depending on two, or three, parameters which does not, generically, preserve meromorphic two-forms. With the introduction of the orbit of the critical set (vanishing condition of the Jacobian), also called ``post-critical set'', we get some new structures, some "non-analytic" two-form which reduce to meromorphic two-forms for particular subvarieties in the parameter space. On these subvarieties, the iterates of the critical set have a polynomial growth in the \emph{degrees of the parameters}, while one has an exponential growth out of these subspaces. The analysis of our birational transformation in $ CP_2$ is first carried out using Diller-Favre criterion in order to find the complexity reduction of the mapping. The integrable cases are found. The identification between the complexity growth and the topological entropy is, one more time, verified. We perform plots of the post-critical set, as well as calculations of Lyapunov exponents for many orbits, confirming that generically no meromorphic two-form can be preserved for this mapping. These birational transformations in $ CP_2$, which, generically, do not preserve any meromorphic two-form, are extremely similar to other birational transformations we previously studied, which do preserve meromorphic two-forms. We note that these two sets of birational transformations exhibit totally similar results as far as topological complexity is concerned, but drastically different results as far as a more ``probabilistic'' approach of dynamical systems is concerned (Lyapunov exponents). With these examples we see that the existence of a preserved meromorphic two-form explains most of the (numerical) discrepancy between the topological and probabilistic approach of dynamical systems.

nlin.CD↗
arXiv2005

Ising model susceptibility: Fuchsian differential equation for $χ^{(4)}$ and its factorization properties

We give the Fuchsian linear differential equation satisfied by $χ^{(4)}$, the ``four-particle'' contribution to the susceptibility of the isotropic square lattice Ising model. This Fuchsian differential equation is deduced from a series expansion method introduced in two previous papers and is applied with some symmetries and tricks specific to $χ^{(4)}$. The corresponding order ten linear differential operator exhibits a large set of factorization properties. Among these factorizations one is highly remarkable: it corresponds to the fact that the two-particle contribution $χ^{(2)}$ is actually a solution of this order ten linear differential operator. This result, together with a similar one for the order seven differential operator corresponding to the three-particle contribution, $χ^{(3)}$, leads us to a conjecture on the structure of all the $ n$-particle contributions $ χ^{(n)}$.

cond-mat.stat-mech↗
arXiv2002

Random Matrix Theory in Lattice Statistical Mechanics

In this short note we collect together known results on the use of Random Matrix Theory in lattice statistical mechanics. The purpose here is two fold. Firstly the RMT analysis provides an intrinsic characterization of integrability, and secondly it appears to be an effective tool to find new integrable models. Various examples from quantum and classical statistical mechanics are presented.

cond-mat.stat-mech↗
arXiv2002

A classification of four-state spin edge Potts models

We classify four-state spin models with interactions along the edges according to their behavior under a specific group of symmetry transformations. This analysis uses the measure of complexity of the action of the symmetries, in the spirit of the study of discrete dynamical systems on the space of parameters of the models, and aims at uncovering solvable ones. We find that the action of these symmetries has low complexity (polynomial growth, zero entropy). We obtain natural parametrizations of various models, among which an unexpected elliptic parametrization of the four-state chiral Potts model, which we use to localize possible integrability conditions associated with high genus curves.

cond-mat.stat-mech↗
arXiv2002

Random Matrix Theory and higher genus integrability: the quantum chiral Potts model

We perform a Random Matrix Theory (RMT) analysis of the quantum four-state chiral Potts chain for different sizes of the chain up to size L=8. Our analysis gives clear evidence of a Gaussian Orthogonal Ensemble statistics, suggesting the existence of a generalized time-reversal invariance. Furthermore a change from the (generic) GOE distribution to a Poisson distribution occurs when the integrability conditions are met. The chiral Potts model is known to correspond to a (star-triangle) integrability associated with curves of genus higher than zero or one. Therefore, the RMT analysis can also be seen as a detector of ``higher genus integrability''.

cond-mat.stat-mech↗
arXiv1999

Real Arnold complexity versus real topological entropy for birational transformations

We consider a family of birational transformations of two variables, depending on one parameter, for which simple rational expressions with integer coefficients, for the exact expression of the dynamical zeta function, have been conjectured. together with an equality between the asymptotic of the Arnold complexity and the exponential of the topological entropy. This identification takes place for the birational mapping seen as a mapping bearing on two complex variables. We revisit this identification between these two quite `universal complexities' by considering now the mapping as a mapping bearing on two real variables. The definitions of the two previous `topological' complexities (Arnold complexity and topological entropy) are modified according to this real-variables point of view. Most of the `universality' is lost, but the identification survives.

chao-dyn↗
arXiv1998

Complexity spectrum of some discrete dynamical systems

We first study birational mappings generated by the composition of the matrix inversion and of a permutation of the entries of $ 3 \times 3 $ matrices. We introduce a semi-numerical analysis which enables to compute the Arnold complexities for all the $9!$ possible birational transformations. These complexities correspond to a spectrum of eighteen algebraic values. We then drastically generalize these results, replacing permutations of the entries by homogeneous polynomial transformations of the entries possibly depending on many parameters. Again it is shown that the associated birational, or even rational, transformations yield algebraic values for their complexities.

chao-dyn↗
arXiv1998

Rational dynamical zeta functions for birational transformations

We propose a conjecture for the exact expression of the dynamical zeta function for a family of birational transformations of two variables, depending on two parameters. This conjectured function is a simple rational expression with integer coefficients. This yields an algebraic value for the topological entropy. Furthermore the generating function for the Arnold complexity is also conjectured to be a rational expression with integer coefficients with the same singularities as for the dynamical zeta function. This leads, at least in this example, to an equality between the Arnold complexity and the exponential of the topological entropy. We also give a semi-numerical method to effectively compute the Arnold complexity.

chao-dyn↗
arXiv1998

Topological entropy and complexity for discrete dynamical systems

To test a possible relation between the topological entropy and the Arnold complexity, and to provide a non trivial example of a rational dynamical zeta function, we introduce a two-parameter family of two-dimensional discrete rational mappings. The generating functions of the number of fixed points, and of the degree of the successive iterates, are both considered. We conjecture rational expressions with integer coefficients for these two generating functions. and a rational expression for the dynamical zeta function. We then deduce algebraic values for the complexity growth and for the exponential of the topological entropy. Moreover, these two numbers happen to be equal for all the values of the parameters. These conjectures are supported by a semi-numerical method we explain. This method also localizes the integrable cases.

chao-dyn↗
arXiv1996

Random Matrix Theory and Classical Statistical Mechanics. I. Vertex Models

A connection between integrability properties and general statistical properties of the spectra of symmetric transfer matrices of the asymmetric eight-vertex model is studied using random matrix theory (eigenvalue spacing distribution and spectral rigidity). For Yang-Baxter integrable cases, including free-fermion solutions, we have found a Poissonian behavior, whereas level repulsion close to the Wigner distribution is found for non-integrable models. For the asymmetric eight-vertex model, however, the level repulsion can also disappearand the Poisson distribution be recovered on (non Yang--Baxter integrable) algebraic varieties, the so-called disorder varieties. We also present an infinite set of algebraic varieties which are stable under the action of an infinite discrete symmetry group of the parameter space. These varieties are possible loci for free parafermions. Using our numerical criterion we have tested the generic calculability of the model on these algebraic varieties.

cond-mat↗