# Entropic repulsion and lack of the $g$-measure property for Dyson models

## TL;DR

This paper demonstrates that Dyson models, a type of long-range Ising model, do not possess the $g$-measure property at low temperatures due to entropic repulsion effects, revealing new insights into their phase transition behavior.

## Contribution

It shows that Gibbs measures for Dyson models in the phase transition region are not $g$-measures, highlighting the impact of entropic repulsion and interface stability in these long-range systems.

## Key findings

- Gibbs measures are not $g$-measures in the phase transition region.
- Entropic repulsion causes the lack of the $g$-measure property.
- Interface stability plays a key role in the behavior of Dyson models.

## Abstract

We consider Dyson models, Ising models with slow polynomial decay, at low temperature and show that its Gibbs measures deep in the phase transition region are not $g$-measures. The main ingredient in the proof is the occurrence of an entropic repulsion effect, which follows from the mesoscopic stability of a (single-point) interface for these long-range models in the phase transition region.