Beyond series expansions: mathematical structures for the susceptibility of the square lattice Ising model

We first study the properties of the Fuchsian ordinary differential equations for the three and four-particle contributions $χ^{(3)}$ and $χ^{(4)}$ of the square lattice Ising model susceptibility. An analysis of some mathematical properties of these Fuchsian differential equations is sketched. For instance, we study the factorization properties of the corresponding linear differential operators, and consider the singularities of the three and four-particle contributions $χ^{(3)}$ and $χ^{(4)}$, versus the singularities of the associated Fuchsian ordinary differential equations, which actually exhibit new ``Landau-like'' singularities. We sketch the analysis of the corresponding differential Galois groups. In particular we provide a simple, but efficient, method to calculate the so-called ``connection matrices'' (between two neighboring singularities) and deduce the singular behaviors of $χ^{(3)}$ and $χ^{(4)}$. We provide a set of comments and speculations on the Fuchsian ordinary differential equations associated with the $n$-particle contributions $χ^{(n)}$ and address the problem of the apparent discrepancy between such a holonomic approach and some scaling results deduced from a Painlevé oriented approach.

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