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# High order Fuchsian equations for the square lattice Ising model: $\tilde{\chi}^{(5)}$

1 ALGORITHMS - Algorithms
Inria Paris-Rocquencourt
Abstract : We consider the Fuchsian linear differential equation obtained (modulo a prime) for $\tilde{\chi}^{(5)}$, 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 $\tilde{\chi}^{(1)}$ and $\tilde{\chi}^{(3)}$ can be removed from $\tilde{\chi}^{(5)}$ 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 $\tilde{\chi}^{(3)}$ and $\tilde{\chi}^{(4)}$. 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 $\tilde{\chi}^{(n)}$'s.
Document type :
Journal articles
Domain :

https://hal.inria.fr/hal-00780426
Contributor : Alin Bostan Connect in order to contact the contributor
Submitted on : Wednesday, January 23, 2013 - 10:10:20 PM
Last modification on : Friday, November 18, 2022 - 10:14:10 AM

### Citation

A. Bostan, S. Boukraa, A. J. Guttmann, S. Hassani, I. Jensen, et al.. High order Fuchsian equations for the square lattice Ising model: $\tilde{\chi}^{(5)}$. Journal of Physics A: Mathematical and Theoretical, 2009, 42 (27), 32pp. ⟨10.1088/1751-8113/42/27/275209⟩. ⟨hal-00780426⟩

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