# The localisation of low-temperature interfaces in $d$ dimensional Ising model

## TL;DR

This paper investigates the behavior of interfaces in the low-temperature Ising model within a finite box, demonstrating that the interface remains close to a certain set of edges with high probability.

## Contribution

It provides a rigorous proof of the localization of low-temperature interfaces in the $d$-dimensional Ising model with Dobrushin boundary conditions.

## Key findings

- Interface is localized within a logarithmic squared distance of certain boundary-connected edges.
- The result holds for general orientations of the box in $bZ^d$.
- The proof advances understanding of phase separation in lattice models.

## Abstract

We study the Ising model in a box $\Lambda$ in $\mathbb{Z}^d$ (not necessarily parallel to the directions of the lattice) with Dobrushin boundary conditions at low temperature. We couple the spin configuration with the configurations under $+$ and $-$ boundary conditions and we define the interface as the edges whose endpoints have the same spins in the $+$ and $-$ configurations but different spins with the Dobrushin boundary conditions. We prove that, inside the box $\Lambda$, the interface is localized within a distance of order $\ln^2|\Lambda|$ of the set of the edges which are connected to the top by a $+$ path and connected to the bottom by a $-$ path.