Phase Transitions on 1d Long-Range Ising Models with Decaying Fields: A Direct Proof via Contours

TL;DR

This paper provides a rigorous contour-based proof of phase transitions in one-dimensional long-range Ising models with polynomial decay, extending previous results to the entire decay region and including cases with external fields.

Contribution

It offers the first direct contour proof of phase transitions for 1D long-range Ising models across all decay rates in (1,2], without requiring large nearest-neighbor interactions.

Findings

Proves phase transition for polynomial decay  in (1,2]

Extends proof to models with decaying external fields

No large nearest-neighbor interaction assumption needed

Abstract

Following seminal work by J. Fr\"ohlich and T. Spencer on the critical exponent $\alpha=2$, we give a proof via contours of phase transition in the one-dimensional long-range ferromagnetic Ising model in the entire region of decay, where phase transition is known to occur, i.e., polynomial decay $\alpha \in (1,2]$. No assumptions that the nearest-neighbor interaction $J(1)$ is large are made. The robustness of the method also yields a proof of phase transition in the presence of a nonsummable external field that decays sufficiently fast.