Phase transitions in low-dimensional long-range random field Ising models

We consider the long-range random field Ising model in dimension $d=1,2$, whereas the long-range interaction is of the form $J_{xy}=\mathrm{\mid }x-y{\mathrm{\mid }}^{-\alpha }$ with $1<\alpha <3\mathrm{/}2$ for $d=1$ and with $2<\alpha \le 3$ for $d=2$. Our main results establish phase transitions in these regimes. In one dimension, we employ a Peierls argument with some novel modification, suitable for dealing with the randomness coming from the external field; in two dimensions, our proof follows that of Affonso, Bissacot, and Maia (2023) with some adaptations, but new ideas are required in the critical case of $\alpha =3$.