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High order Fuchsian equations for the square lattice Ising model: ${\stackrel{˜}{\chi }}^{\left(5\right)}$. (English) Zbl 1171.82005
Summary: We consider the Fuchsian linear differential equation obtained (modulo a prime) for ${\stackrel{˜}{\chi }}^{\left(5\right)}$, the five-particle contribution to the susceptibility of the square lattice Ising model. We show that one can understand the factorization of the corresponding linear differential operator from calculations using just a single prime. A particular linear combination of ${\stackrel{˜}{\chi }}^{\left(1\right)}$ and ${\stackrel{˜}{\chi }}^{\left(3\right)}$ can be removed from ${\stackrel{˜}{\chi }}^{\left(5\right)}$ and the resulting series is annihilated by a high order globally nilpotent linear ODE. The corresponding (minimal order) linear differential operator, of order 29, splits into factors of small orders. A fifth-order linear differential operator occurs as the left-most factor of the ‘depleted’ differential operator and it is shown to be equivalent to the symmetric fourth power of ${L}_{E}$, the linear differential operator corresponding to the elliptic integral $E$. This result generalizes what we have found for the lower order terms ${\stackrel{˜}{\chi }}^{\left(3\right)}$ and ${\stackrel{˜}{\chi }}^{\left(4\right)}$. We conjecture that a linear differential operator equivalent to a symmetric $\left(n-1\right)$ th power of ${L}_{E}$ occurs as a left-most factor in the minimal order linear differential operators for all ${\stackrel{˜}{\chi }}^{\left(n\right)}$’s.
##### MSC:
 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 34M55 Painlevé and other special equations; classification, hierarchies 47E05 Ordinary differential operators 81Qxx General mathematical topics and methods in quantum theory 32G34 Moduli and deformations for ordinary differential equations 34Lxx Ordinary differential operators 34Mxx Differential equations in the complex domain 14Kxx Abelian varieties and schemes
##### Keywords:
Fuchsin equations; Ising model