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Using a single functional form which is able to represent several differentclasses of statistical distributions, we introduce a preliminary study of theferromagnetic Ising model on the cubic lattices under the influence ofnon-Gaussian local external magnetic field. Specifically, depending on thevalue of the tail parameter, $\tau $ ($\tau < 3$), we assign a quenched randomfield that can be platykurtic (sub-Gaussian) or leptokurtic (fat-tailed) form.For $\tau< 5/3$, such distributions have finite standard deviation and they areeither the Student-$t$ ($1< \tau< 5/3$) or the $r$-distribution ($\tau< 1$)extended to all plausible real degrees of freedom with the Gaussian beingretrieved in the limit $\tau \rightarrow 1$. Otherwise, the distribution hasgot the same asymptotic power-law behaviour as the $\alpha$-stable L\'{e}vydistribution with $\alpha = (3 - \tau)/(\tau - 1)$. The uniform distribution isachieved in the limit $\tau \rightarrow \infty$. Our results purport theexistence of ferromagnetic order at finite temperatures for all the studiedvalues of $\tau$ with some mean-field predictions surviving in thethree-dimensional case.

Fat-tailed and compact random-field Ising models on cubic lattices