We give the Fuchsian linear differential equation satisfied by χ(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 χ(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 χ(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, χ(3), leads us to a conjecture on the structure of all the n-particle contributions χ(n).
@article{arxiv.cond-mat/0502155,
title = {Ising model susceptibility: Fuchsian differential equation for $\chi^{(4)}$ and its factorization properties},
author = {N. Zenine and S. Boukraa and S. Hassani and J. -M. Maillard},
journal= {arXiv preprint arXiv:cond-mat/0502155},
year = {2009}
}