A Bostan et al 2009 J. Phys. A: Math. Theor. 42 275209 doi:10.1088/1751-8113/42/27/275209
High order Fuchsian equations for the square lattice Ising model: 
A Bostan1, S Boukraa2, A J Guttmann3, S Hassani4, I Jensen3, J-M Maillard5 and N Zenine4
Show affiliationsWe consider the Fuchsian linear differential equation obtained (modulo a prime) for 
, 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 
and 
can be removed from 
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 LE, the linear differential operator corresponding to the elliptic integral E. This result generalizes what we have found for the lower order terms 
and 
. We conjecture that a linear differential operator equivalent to a symmetric (n − 1) th power of LE occurs as a left-most factor in the minimal order linear differential operators for all 
's.
- PACS
-
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
75.30.Cr Saturation moments and magnetic susceptibilities
02.30.Hq Ordinary differential equations
- MSC
-
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
47E05 Ordinary differential operators (See also 34Bxx, 34Lxx)
34Lxx Ordinary differential operators (See also 47E05)
47A68 Factorization theory (including Wiener-Hopf and spectral factorizations)
- Subjects
- Dates
-
Issue 27 (10 July 2009)
Received 9 April 2009, in final form 16 May 2009
Published 17 June 2009
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High order Fuchsian equations for the square lattice Ising model:
A Bostan et al 2009 J. Phys. A: Math. Theor. 42 275209
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