The Fuchsian differential equation of the square lattice Ising model  ⁽³⁾susceptibility

Landau singularities and singularities of holonomic integrals of the Ising class

“…One does remark, on this second example, that the singularities of the linear ODE for χ (3) [4,5] actually pop out in the linear ODE of this "toy integral", in particular the quite non trivial 1 + 3 w + 4 w 2 = 0 singularities. The linear differential equation (41) is of Fuchsian type.…”

“…This connection matrix method [8] was used to find the singular behavior of χ (3) and χ (4) , the third and fourth "particle" contribution to the Ising magnetic susceptibility [8]. Let us sketch the method.…”

“…These singularities emerge as a consequence of inclusion of the y i . Recall that these last singularities do not occur [4] in the linear ODE corresponding to χ (3) . Do they disappear in χ (3) as a consequence of the enlargment of multidimensional integration domain, or as a consequence of the inclusion of the G (n) factor ?…”

“…The linear ODE corresponding to χ (4) has [6], besides the known singularities s 2 = 1, and s 2 = −1, only the singularities (1) predicted by Nickel [1, 2], while the linear ODE corresponding to χ (3) has shown a pair of singularities 1 + 3w + 4w 2 = 0 (where w = s/2/(1 + s 2 )) that are not on the unit circle |s| = 1. The condition |s| = 1 is equivalent, for the variable w, to be real and such that its absolute value is greater than 1/4.…”

“…The integrands of this class of integrals have a trigonometric-hyperbolic form and are simple algebraic functions in the cosinus of the angular integration variables, and are also algebraic in the remaining w variable. These integrals are thus holonomic: they must satisfy finite order linear ODE's with polynomial coefficients in the remaining variable w. The route we take for obtaining the linear ODE's of the two sets of integrals will be through series expansions [4,5,6].…”