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The Fuchsian differential equation of the square lattice Ising model ${\chi }^{\left(3\right)}$ susceptibility. (English) Zbl 1073.82014

Summary: Using an expansion method in the variables ${x}_{i}$ that appear in $\left(n-1\right)$-dimensional integrals representing the $n$-particle contribution to the Ising square lattice model susceptibility $\chi$, we generate a long series of coefficients for the three-particle contribution ${\chi }^{\left(3\right)}$, using an ${N}^{4}$ polynomial time algorithm. We give the Fuchsian differential equation of order 7 for ${\chi }^{\left(3\right)}$ that reproduces all the terms of our long series. An analysis of the properties of this Fuchsian differential equation is performed.

For additional information see [B. G. Nickel, ibid. 38, No. 20, 4517–4518 (2005; Zbl 1074.82010).

##### MSC:
 82B27 Critical phenomena (equilibrium statistical mechanics) 34M55 Painlevé and other special equations; classification, hierarchies 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs