Selected non-holonomic functions in lattice statistical mechanics and enumerative combinatorics

We recall that the full susceptibility series of the Ising model, modulo powers of the prime 2, reduce to algebraic functions. We also recall the non-linear polynomial differential equation obtained by Tutte for the generating function of the q-coloured rooted triangulations by vertices, which is known to have algebraic solutions for all the numbers of the form $2 +2 \cos(j\pi/n)$, the holonomic status of the q= 4 being unclear. We focus on the analysis of the q= 4 case, showing that the corresponding series is quite certainly non-holonomic. Along the line of a previous work on the susceptibility of the Ising model, we consider this q=4 series modulo the first eight primes 2, 3, ... 19, and show that this (probably non-holonomic) function reduces, modulo these primes, to algebraic functions. We conjecture that this probably non-holonomic function reduces to algebraic functions modulo (almost) every prime, or power of prime numbers. This raises the question to see whether such remarkable non-holonomic functions can be seen as ratio of diagonals of rational functions, or algebraic, functions of diagonals of rational functions.Comment: 27 page

Scaling functions in the square Ising model

We show and give the linear differential operators ${\cal L}^{scal}_q$ of order q= n^2/4+n+7/8+(-1)^n/8, for the integrals $I_n(r)$ which appear in the two-point correlation scaling function of Ising model $F_{\pm}(r)= \lim_{scaling} {\cal M}_{\pm}^{-2} = \sum_{n} I_{n}(r)$. The integrals $I_{n}(r)$ are given in expansion around r= 0 in the basis of the formal solutions of $\, {\cal L}^{scal}_q$ with transcendental combination coefficients. We find that the expression $r^{1/4}\,\exp(r^2/8)$ is a solution of the Painlev\'e VI equation in the scaling limit. Combinations of the (analytic at $r= 0$) solutions of ${\cal L}^{scal}_q$ sum to $\exp(r^2/8)$. We show that the expression $r^{1/4} \exp(r^2/8)$ is the scaling limit of the correlation function $C(N, N)$ and $C(N, N+1)$. The differential Galois groups of the factors occurring in the operators ${\cal L}^{scal}_q$ are given.Comment: 26 page

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

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 $\chi^{(3)}$ at the points $1+3w+4w^2= 0$ and show that $\chi^{(3)}(s)$ 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 ($| s | > 1$).Comment: 34 pages, 1 figur

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 $\chi^{(3)}$ 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.Comment: 55 page

Painleve versus Fuchs

The sigma form of the Painlev{\'e} 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{\'e} 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{\'e}.Comment: 18 pages, Dedicated to the centenary of the publication of the Painleve VI equation in the Comptes Rendus de l'Academie des Sciences de Paris by Richard Fuchs in 190

Diagonal Ising susceptibility: elliptic integrals, modular forms and Calabi-Yau equations

We give the exact expressions of the partial susceptibilities $\chi^{(3)}_d$ and $\chi^{(4)}_d$ for the diagonal susceptibility of the Ising model in terms of modular forms and Calabi-Yau ODEs, and more specifically, $_3F_2([1/3,2/3,3/2],\, [1,1];\, z)$ and $_4F_3([1/2,1/2,1/2,1/2],\, [1,1,1]; \, z)$ hypergeometric functions. By solving the connection problems we analytically compute the behavior at all finite singular points for $\chi^{(3)}_d$ and $\chi^{(4)}_d$. We also give new results for $\chi^{(5)}_d$. We see in particular, the emergence of a remarkable order-six operator, which is such that its symmetric square has a rational solution. These new exact results indicate that the linear differential operators occurring in the $n$-fold integrals of the Ising model are not only "Derived from Geometry" (globally nilpotent), but actually correspond to "Special Geometry" (homomorphic to their formal adjoint). This raises the question of seeing if these "special geometry" Ising-operators, are "special" ones, reducing, in fact systematically, to (selected, k-balanced, ...) $_{q+1}F_q$ hypergeometric functions, or correspond to the more general solutions of Calabi-Yau equations.Comment: 35 page