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

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

This paper deals with $\tilde{\chi}^{(6)}$, the six-particle contribution to the magnetic susceptibility of the square lattice Ising model. We have generated, modulo a prime, series coefficients for $\tilde{\chi}^{(6)}$. 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 $\Phi^{(6)}=\tilde{\chi}^{(6)} - {2 \over 3} \tilde{\chi}^{(4)} + {2 \over 45} \tilde{\chi}^{(2)}$. 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 $\tilde{\chi}^{(5)}$. 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.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 $\chi^{(n)}$ of the magnetic susceptibility of the Ising model ($n \le 6$) 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 $\chi^{(5)}$ and $\chi^{(6)}$), 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 $Sp(12, \mathbb{C})$. 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 $SO(21, \mathbb{C})$.Comment: 33 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 $q \times q$ matrices: the inversion of the $q \times q$ 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 $q=4$, 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

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

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

We calculate very long low- and high-temperature series for the susceptibility $\chi$ of the square lattice Ising model as well as very long series for the five-particle contribution $\chi^{(5)}$ and six-particle contribution $\chi^{(6)}$. 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 $\chi^{(5)}$ 10000 terms of the series are calculated {\it modulo} a single prime, and have been used to find the linear ODE satisfied by $\chi^{(5)}$ {\it modulo} a prime. A diff-Pad\'e analysis of 2000 terms series for $\chi^{(5)}$ and $\chi^{(6)}$ 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 $\chi^{(5)}$, and $w^2= 1/8$ for the ODE of $\chi^{(6)}$, which are {\it not} singularities of the ``physical'' $\chi^{(5)}$ and $\chi^{(6)},$ that is to say the series-solutions of the ODE's which are analytic at $w =0$. Furthermore, analysis of the long series for $\chi^{(5)}$ (and $\chi^{(6)}$) combined with the corresponding long series for the full susceptibility $\chi$ yields previously conjectured singularities in some $\chi^{(n)}$, $n \ge 7$. We also present a mechanism of resummation of the logarithmic singularities of the $\chi^{(n)}$ leading to the known power-law critical behaviour occurring in the full $\chi$, and perform a power spectrum analysis giving strong arguments in favor of the existence of a natural boundary for the full susceptibility $\chi$.Comment: 54 pages, 2 figure