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Ising model susceptibility: the Fuchsian differential equation for $\chi^{(4)}$ and its factorization properties. (English) Zbl 1086.82520
Summary: 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 $\chi^{(n)}$.

##### MSC:
 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 34M55 Painlevé and other special equations; classification, hierarchies 47E05 Ordinary differential operators 81Q05 Closed and approximate solutions to quantum-mechanical equations 32G34 Moduli and deformations for ordinary differential equations
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