Exponential decay of loop lengths in the loop $O(n)$ model with large $n$

TL;DR

This paper proves that in the loop $O(n)$ model on the hexagonal lattice with large $n$, long loops decay exponentially, and describes the phase transition between dilute disordered and dense ordered phases based on parameters.

Contribution

It establishes exponential decay of correlations for large $n$ in the loop $O(n)$ model and characterizes the phase transition depending on the parameters.

Findings

Long loops are exponentially unlikely for large $n$

Existence of dilute and dense phases depending on parameters

Characterization of typical configurations in different phases

Abstract

The loop $O(n)$ model is a model for a random collection of non-intersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin $O(n)$ model. It has been conjectured that both the spin and the loop $O(n)$ models exhibit exponential decay of correlations when $n>2$. We verify this for the loop $O(n)$ model with large parameter $n$, showing that long loops are exponentially unlikely to occur, uniformly in the edge weight $x$. Our proof provides further detail on the structure of typical configurations in this regime. Putting appropriate boundary conditions, when $nx^6$ is sufficiently small, the model is in a dilute, disordered phase in which each vertex is unlikely to be surrounded by any loops, whereas when $nx^6$ is sufficiently large, the model is in a dense, ordered phase which is a small perturbation of one of the three ground states.