152 research outputs found
High order Fuchsian equations for the square lattice Ising model:
This paper deals with , the six-particle contribution to
the magnetic susceptibility of the square lattice Ising model. We have
generated, modulo a prime, series coefficients for . 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
. 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 . 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 . 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.Comment: 23 page
The Ising model and Special Geometries
We show that the globally nilpotent G-operators corresponding to the factors
of the linear differential operators annihilating the multifold integrals
of the magnetic susceptibility of the Ising model () are
homomorphic to their adjoint. This property of being self-adjoint up to
operator homomorphisms, is equivalent to the fact that their symmetric square,
or their exterior square, have rational solutions. The differential Galois
groups are in the special orthogonal, or symplectic, groups. This self-adjoint
(up to operator equivalence) property means that the factor operators we
already know to be Derived from Geometry, are special globally nilpotent
operators: they correspond to "Special Geometries".
Beyond the small order factor operators (occurring in the linear differential
operators associated with and ), and, in particular,
those associated with modular forms, we focus on the quite large order-twelve
and order-23 operators. We show that the order-twelve operator has an exterior
square which annihilates a rational solution. Then, its differential Galois
group is in the symplectic group . The order-23 operator
is shown to factorize in an order-two operator and an order-21 operator. The
symmetric square of this order-21 operator has a rational solution. Its
differential Galois group is, thus, in the orthogonal group
.Comment: 33 page
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 , with , where is the
usual Ising model coupling constant. Switching to the variable , 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 at the
points and show that is not singular at the
corresponding points inside the unit circle , while its analytical
continuation in the variable is actually singular at the corresponding
points oustside the unit circle ().Comment: 34 pages, 1 figur
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
and of the corresponding susceptibility, and obtain linear
differential equations for the three and four particle contributions, as well
as the five particle contribution , 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 -particle contributions are
quite directly inherited from the direct sum structure on the form factors .
We show that the particle contributions have their
singularities at roots of unity. These singularities become dense on the unit
circle as .Comment: 18 page
Canonical decomposition of linear differential operators with selected differential Galois groups
We revisit an order-six linear differential operator having a solution which
is a diagonal of a rational function of three variables. Its exterior square
has a rational solution, indicating that it has a selected differential Galois
group, and is actually homomorphic to its adjoint. We obtain the two
corresponding intertwiners giving this homomorphism to the adjoint. We show
that these intertwiners are also homomorphic to their adjoint and have a simple
decomposition, already underlined in a previous paper, in terms of order-two
self-adjoint operators. From these results, we deduce a new form of
decomposition of operators for this selected order-six linear differential
operator in terms of three order-two self-adjoint operators. We then generalize
the previous decomposition to decompositions in terms of an arbitrary number of
self-adjoint operators of the same parity order. This yields an infinite family
of linear differential operators homomorphic to their adjoint, and, thus, with
a selected differential Galois group. We show that the equivalence of such
operators is compatible with these canonical decompositions. The rational
solutions of the symmetric, or exterior, squares of these selected operators
are, noticeably, seen to depend only on the rightmost self-adjoint operator in
the decomposition. These results, and tools, are applied on operators of large
orders. For instance, it is seen that a large set of (quite massive) operators,
associated with reflexive 4-polytopes defining Calabi-Yau 3-folds, obtained
recently by P. Lairez, correspond to a particular form of the decomposition
detailed in this paper.Comment: 40 page
Integrable mappings and polynomial growth
We describe birational representations of discrete groups generated by
involutions, having their origin in the theory of exactly solvable
vertex-models in lattice statistical mechanics. These involutions correspond
respectively to two kinds of transformations on matrices: the
inversion of the matrix and an (involutive) permutation of the
entries of the matrix. We concentrate on the case where these permutations are
elementary transpositions of two entries. In this case the birational
transformations fall into six different classes. For each class we analyze the
factorization properties of the iteration of these transformations. These
factorization properties enable to define some canonical homogeneous
polynomials associated with these factorization properties. Some mappings yield
a polynomial growth of the complexity of the iterations. For three classes the
successive iterates, for , actually lie on elliptic curves. This analysis
also provides examples of integrable mappings in arbitrary dimension, even
infinite. Moreover, for two classes, the homogeneous polynomials are shown to
satisfy non trivial non-linear recurrences. The relations between
factorizations of the iterations, the existence of recurrences on one or
several variables, as well as the integrability of the mappings are analyzed.Comment: 45 page
Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity
We show that the n-fold integrals 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.Comment: 100 page
Ising n-fold integrals as diagonals of rational functions and integrality of series expansions
We show that the n-fold integrals 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.Comment: This paper is the short version of the larger (100 pages) version,
available as arXiv:1211.6031 , where all the detailed proofs are given and
where a much larger set of examples is displaye
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