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Ising model susceptibility: the Fuchsian differential equation for χ (4) and its factorization properties. (English) Zbl 1086.82520
Summary: 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) .
MSC:
82B20Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
34M55Painlevé and other special equations; classification, hierarchies
47E05Ordinary differential operators
81Q05Closed and approximate solutions to quantum-mechanical equations
32G34Moduli and deformations for ordinary differential equations