We present a simple, but efficient, way to calculate connection matrices
between sets of independent local solutions, defined at two neighboring
singular points, of Fuchsian differential equations of quite large orders, such
as those found for the third and fourth contribution ($\chi^{(3)}$ and
$\chi^{(4)}$) to the magnetic susceptibility of square lattice Ising model. We
deduce all the critical behaviors of the solutions $\chi^{(3)}$ and
$\chi^{(4)}$, as well as the asymptotic behavior of the coefficients in the
corresponding series expansions. We confirm that the newly found quadratic
number singularities of the Fuchsian ODE associated to $\chi^{(3)}$ are not
singularities of the particular solution $\chi^{(3)}$ itself.
We use the previous connection matrices to get the exact expressions of all
the monodromy matrices of the Fuchsian differential equation for $\chi^{(3)}$
(and $\chi^{(4)}$) expressed in the same basis of solutions. These monodromy
matrices are the generators of the differential Galois group of the Fuchsian
differential equations for $\chi^{(3)}$ (and $\chi^{(4)}$), whose analysis is
just sketched here.
As far as the physics implications of the solutions are concerned, we find
challenging qualitative differences when comparing the corrections to scaling
for the full susceptibillity $\chi$ at high temperature (resp. low temperature)
and the first two terms $\chi^{(1)}$ and $\chi^{(3)}$
(resp. $\chi^{(2)}$ and $\chi^{(4)}$) .