# Markov chain sampling of the $O(n)$ loop models on the infinite plane

## TL;DR

This paper extends a numerical Markov chain sampling method to the $O(n)$ loop models on the infinite plane, demonstrating that scale invariance enables efficient sampling similar to the Ising model, despite non-local Gibbs measures.

## Contribution

It introduces an extension of the Markov chain sampling method to $O(n)$ models, showing the approach's effectiveness beyond the Ising case for scale-invariant systems.

## Key findings

- Efficient sampling of $O(n)$ models on the infinite plane is possible.
- Scale invariance is sufficient for the numerical method to work.
- The method's efficiency is comparable to that for the Ising model.

## Abstract

It was recently proposed in https://journals.aps.org/pre/abstract/10.1103/PhysRevE.94.043322 [Herdeiro & Doyon Phys.,Rev.,E (2016)] a numerical method showing a precise sampling of the infinite plane 2d critical Ising model for finite lattice subsections. The present note extends the method to a larger class of models, namely the $O(n)$ loop gas models for $n \in (1,2]$. We argue that even though the Gibbs measure is non local, it is factorizable on finite subsections when sufficient information on the loops touching the boundaries is stored. Our results attempt to show that provided an efficient Markov chain mixing algorithm and an improved discrete lattice dilation procedure the planar limit of the $O(n)$ models can be numerically studied with efficiency similar to the Ising case. This confirms that scale invariance is the only requirement for the present numerical method to work.