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

Abstract

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 ${\Phi }^{(6)}={\tilde{\chi }}^{(6)}-\frac{2}{3}{\tilde{\chi }}^{(4)}+\frac{2}{45}{\tilde{\chi }}^{(2)}$. 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.

23 pages

References in corpus (3)

• Experimental mathematics on the magnetic susceptibility of the square lattice Ising model
• Singularities of $n$-fold integrals of the Ising class and the theory of elliptic curves
• Globally nilpotent differential operators and the square Ising model

Cited by in corpus (18)

• Ising n-fold integrals as diagonals of rational functions and integrality of series expansions
• The Ising model: from elliptic curves to modular forms and Calabi-Yau equations
• A fast algorithm for computing the characteristic polynomial of the p-curvature
• On Christol's conjecture
• Lattice Green Functions: the seven-dimensional face-centred cubic lattice
• Holonomic functions of several complex variables and singularities of anisotropic Ising n-fold integrals
• Computation of the Similarity Class of the p-Curvature
• Diagonals of rational functions, pullbacked 2F1 hypergeometric functions and modular forms (unabrigded version)
• Selected non-holonomic functions in lattice statistical mechanics and enumerative combinatorics
• Canonical decomposition of linear differential operators with selected differential Galois groups
• Is the full susceptibility of the square-lattice Ising model a differentially algebraic function?
• Scaling functions in the square Ising model
• The Ising model and Special Geometries
• The lambda extensions of the Ising correlation functions C(M, N)
• Factorization of the Ising model form factors
• Magnetic susceptibility of the square lattice Ising model
• On General-n Coefficients in Series Expansions for Row Spin-Spin Correlation Functions in the Two-Dimensional Ising Model
• On the Singularities in the Susceptibility Expansion for the Two-Dimensional Ising Model