Ising model susceptibility: Fuchsian differential equation for $\\chi^{(4)}$ and its factorization properties
Résumé
We give the Fuchsian linear differential equation satisfied by $\\chi^{(4)}$, the ``four-particle\'\' contribution to the susceptibility of the isotropic square lattice Ising model. This Fuchsian differential equation is deduced from a series expansion method introduced in two previous papers and is applied with some symmetries and tricks specific to $\\chi^{(4)}$. The corresponding order ten linear differential operator exhibits a large set of factorization properties. Among these factorizations one is highly remarkable: it corresponds to the fact that the two-particle contribution $\\chi^{(2)}$ is actually a solution of this order ten linear differential operator. This result, together with a similar one for the order seven differential operator corresponding to the three-particle contribution, $\\chi^{(3)}$, leads us to a conjecture on the structure of all the $ n$-particle contributions $ \\chi^{(n)}$.