2,537 research outputs found
Diagonals of rational functions, pullbacked 2F1 hypergeometric functions and modular forms (unabrigded version)
We recall that diagonals of rational functions naturally occur in lattice
statistical mechanics and enumerative combinatorics. We find that a
seven-parameter rational function of three variables with a numerator equal to
one (reciprocal of a polynomial of degree two at most) can be expressed as a
pullbacked 2F1 hypergeometric function. This result can be seen as the simplest
non-trivial family of diagonals of rational functions. We focus on some
subcases such that the diagonals of the corresponding rational functions can be
written as a pullbacked 2F1 hypergeometric function with two possible rational
functions pullbacks algebraically related by modular equations, thus showing
explicitely that the diagonal is a modular form. We then generalise this result
to eight, nine and ten parameters families adding some selected cubic terms at
the denominator of the rational function defining the diagonal. We finally show
that each of these previous rational functions yields an infinite number of
rational functions whose diagonals are also pullbacked 2F1 hypergeometric
functions and modular forms.Comment: 39 page
Holonomic functions of several complex variables and singularities of anisotropic Ising n-fold integrals
Lattice statistical mechanics, often provides a natural (holonomic) framework
to perform singularity analysis with several complex variables that would, in a
general mathematical framework, be too complex, or could not be defined.
Considering several Picard-Fuchs systems of two-variables "above" Calabi-Yau
ODEs, associated with double hypergeometric series, we show that holonomic
functions are actually a good framework for actually finding the singular
manifolds. We, then, analyse the singular algebraic varieties of the n-fold
integrals , corresponding to the decomposition of the magnetic
susceptibility of the anisotropic square Ising model. We revisit a set of
Nickelian singularities that turns out to be a two-parameter family of elliptic
curves. We then find a first set of non-Nickelian singularities for and , that also turns out to be rational or ellipic
curves. We underline the fact that these singular curves depend on the
anisotropy of the Ising model. We address, from a birational viewpoint, the
emergence of families of elliptic curves, and of Calabi-Yau manifolds on such
problems. We discuss the accumulation of these singular curves for the
non-holonomic anisotropic full susceptibility.Comment: 36 page
A comment on free-fermion conditions for lattice models in two and more dimensions
We analyze free-fermion conditions on vertex models. We show --by examining
examples of vertex models on square, triangular, and cubic lattices-- how they
amount to degeneration conditions for known symmetries of the Boltzmann
weights, and propose a general scheme for such a process in two and more
dimensions.Comment: 12 pages, plain Late
On Christol's conjecture
We show that the unresolved examples of Christol's conjecture
\, _3F_{2}\left([2/9,5/9,8/9],[2/3,1],x\right) and
, are indeed diagonals of
rational functions. We also show that other \, _3F_2 and \, _4F_3
unresolved examples of Christol's conjecture are diagonals of rational
functions. Finally we give two arguments that show that it is likely that the
\, _3F_2([1/9, 4/9, 5/9], \, [1/3,1], \, 27 \cdot x) function is a diagonal
of a rational function.Comment: 13 page
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.Comment: 25 pages, 7 PostScript Figure
Factorization of Ising correlations C(M,N) for and M+N odd, , and their lambda extensions
We study the factorizations of Ising low-temperature correlations C(M,N) for
and M+N odd, , for both the cases where there are
two factors, and where there are four factors. We find that the two
factors for satisfy the same non-linear differential equation and,
similarly, for M=0 the four factors each satisfy Okamoto sigma-form of
Painlev\'e VI equations with the same Okamoto parameters. Using a Landen
transformation we show, for , that the previous non-linear
differential equation can actually be reduced to an Okamoto sigma-form of
Painlev\'e 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 Painlev\'e 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 Painlev\'e 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 (Painlev\'e V) relations between the sum and
difference of sigma's to this Painlev\'e VI framework.Comment: 45 page
Heun functions and diagonals of rational functions (unabridged version)
International audienceWe provide a set of diagonals of simple rational functions of four variables that are seen to be squares of Heun functions. Each time, these Heun functions, obtained by creative telescoping, turn out to be pullbacked 2 F 1 hypergeometric functions and in fact classical modular forms. We even obtained Heun functions that are automorphic forms associated with Shimura curves as solutions of telescopers of rational functions
Diagonals of rational functions, pullbacked 2 F 1 hypergeometric functions and modular forms (unabrigded version)
International audienceWe recall that diagonals of rational functions naturally occur in lattice statistical mechanics and enumerative combinatorics. We find that a seven-parameter rational function of three variables with a numerator equal to one (reciprocal of a polynomial of degree two at most) can be expressed as a pullbacked 2 F 1 hypergeometric function. This result can be seen as the simplest non-trivial family of diagonals of rational functions. We focus on some subcases such that the diagonals of the corresponding rational functions can be written as a pullbacked 2 F 1 hypergeometric function with two possible rational functions pullbacks algebraically related by modular equations, thus showing explicitely that the diagonal is a modular form. We then generalise this result to eight, nine and ten parameters families adding some selected cubic terms at the denominator of the rational function defining the diagonal. We finally show that each of these previous rational functions yields an infinite number of rational functions whose diagonals are also pullbacked 2 F 1 hypergeometric functions and modular forms
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