Square lattice Ising model (chi)over-tilde((5)) ODE in exact arithmetic

Abstract

We obtain in exact arithmetic the order 24 linear differential operator L24 and the right-hand side E(5) of the inhomogeneous equation L24(Φ(5)) = E(5), where φ(5) = χ̃(5) - χ̃(3)/2 + χ̃(1)/120 is a linear combination of n-particle contributions to the susceptibility of the square lattice Ising model. In Bostan et al (2009 J. Phys. A: Math. Theor. 42 275209), the operator L24 (modulo a prime) was shown to factorize into L(left)12 · L(right)12; here we prove that no further factorization of the order 12 operator L(left)12 is possible. We use the exact ODE to obtain the behaviour of χ̃ (5) at the ferromagnetic critical point and to obtain a limited number of analytic continuations of χ̃(5) bey..