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High order Fuchsian equations for the square lattice Ising model: $\widetilde{\chi}^{(5)}$. (English) Zbl 1171.82005
Summary: We consider the Fuchsian linear differential equation obtained (modulo a prime) for $\widetilde{\chi}^{(5)}$, 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 $\widetilde{\chi}^{(1)}$ and $\widetilde{\chi}^{(3)}$ can be removed from $\widetilde{\chi}^{(5)}$ 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 $\widetilde{\chi}^{(3)}$ and $\widetilde{\chi}^{(4)}$. We conjecture that a linear differential operator equivalent to a symmetric $(n - 1)$ th power of $L_{E}$ occurs as a left-most factor in the minimal order linear differential operators for all $\widetilde{\chi}^{(n)}$’s.

82B20Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
34M55Painlevé and other special equations; classification, hierarchies
47E05Ordinary differential operators
81QxxGeneral mathematical topics and methods in quantum theory
32G34Moduli and deformations for ordinary differential equations
34LxxOrdinary differential operators
34MxxDifferential equations in the complex domain
14KxxAbelian varieties and schemes
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