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

Abstract

We consider the long-range random field Ising model in dimension $d=1,2$, whereas the long-range interaction is of the form $J_{xy}=|x-y{|}^{-\hat{I}\pm }$ with $1<\hat{I}\pm <3/2$ for $d=1$ and with $2<\hat{I}\pm \le 3$ 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 $\hat{I}\pm =3$.

42 pages, 7 figures