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 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 $\chi_{d}^{(1)}$ and $\chi_{d}^{(2)}$ of the corresponding susceptibility, and obtain linear differential equations for the three and four particle contributions, as well as the five particle contribution ${\chi}^{(5)}_d(t)$, 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 $n$-particle contributions $\chi_{d}^{(n)}$ are quite directly inherited from the direct sum structure on the form factors $f^{(n)}$. We show that the $n^{th}$ particle contributions $\chi_{d}^{(n)}$ have their singularities at roots of unity. These singularities become dense on the unit circle $|\sinh2E_v/kT \sinh 2E_h/kT|=1$ as $n\to \infty$.Comment: 18 page

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

High order Fuchsian equations for the square lattice Ising model: χ~(5)\tilde{\chi}^{(5)}

33 pagesInternational audienceWe consider the Fuchsian linear differential equation obtained (modulo a prime) for $\tilde{\chi}^{(5)}$, the five-particle contribution to the susceptibility of the square lattice Ising model. We show that one can understand the factorization of the corresponding linear differential operator from calculations using just a single prime. A particular linear combination of $\tilde{\chi}^{(1)}$ and $\tilde{\chi}^{(3)}$ can be removed from $\tilde{\chi}^{(5)}$ and the resulting series is annihilated by a high order globally nilpotent linear ODE. The corresponding (minimal order) linear differential operator, of order 29, splits into factors of small orders. A fifth order linear differential operator occurs as the left-most factor of the "depleted" differential operator and it is shown to be equivalent to the symmetric fourth power of $L_E$, the linear differential operator corresponding to the elliptic integral $E$. This result generalizes what we have found for the lower order terms $\tilde{\chi}^{(3)}$ and $\tilde{\chi}^{(4)}$. We conjecture that a linear differential operator equivalent to a symmetric $(n-1)$-th power of $L_E$ occurs as a left-most factor in the minimal order linear differential operators for all $\tilde{\chi}^{(n)}$'s

Lattice Green functions: the d -dimensional face-centered cubic lattice, d = 8, 9, 10, 11, 12

International audienc