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

TL;DR

This paper investigates phase transitions in low-dimensional long-range random field Ising models with specific decay exponents, employing novel and adapted methods to establish the existence of phase transitions in these regimes.

Contribution

The paper proves phase transitions in 1D and 2D long-range RFIMs for certain decay exponents, introducing new modifications to existing proofs and addressing critical cases.

Findings

Phase transitions are established for 1D models with $1< \alpha < 3/2$.

Phase transitions are established for 2D models with $2 < \alpha \,\leq 3$.

Novel modifications to Peierls argument and proof techniques are introduced.

Abstract

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