  Using a single functional form which is able to represent several different
classes of statistical distributions, we introduce a preliminary study of the
ferromagnetic Ising model on the cubic lattices under the influence of
non-Gaussian local external magnetic field. Specifically, depending on the
value of the tail parameter, $\tau $ ($\tau < 3$), we assign a quenched random
field that can be platykurtic (sub-Gaussian) or leptokurtic (fat-tailed) form.
For $\tau< 5/3$, such distributions have finite standard deviation and they are
either the Student-$t$ ($1< \tau< 5/3$) or the $r$-distribution ($\tau< 1$)
extended to all plausible real degrees of freedom with the Gaussian being
retrieved in the limit $\tau \rightarrow 1$. Otherwise, the distribution has
got the same asymptotic power-law behaviour as the $\alpha$-stable L\'{e}vy
distribution with $\alpha = (3 - \tau)/(\tau - 1)$. The uniform distribution is
achieved in the limit $\tau \rightarrow \infty$. Our results purport the
existence of ferromagnetic order at finite temperatures for all the studied
values of $\tau$ with some mean-field predictions surviving in the
three-dimensional case.
