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

Abstract

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, $τ$ ($τ< 3$), we assign a quenched random field that can be platykurtic (sub-Gaussian) or leptokurtic (fat-tailed) form. For $τ< 5/3$, such distributions have finite standard deviation and they are either the Student-$t$ ($1< τ< 5/3$) or the $r$-distribution ($τ< 1$) extended to all plausible real degrees of freedom with the Gaussian being retrieved in the limit $τ\rightarrow 1$. Otherwise, the distribution has got the same asymptotic power-law behaviour as the $α$-stable Lévy distribution with $α= (3 - τ)/(τ- 1)$. The uniform distribution is achieved in the limit $τ\rightarrow \infty$. Our results purport the existence of ferromagnetic order at finite temperatures for all the studied values of $τ$ with some mean-field predictions surviving in the three-dimensional case.