Exponential decay in the loop $O(n)$ model: $n> 1$, $x<\tfrac{1}{\sqrt{3}}+\varepsilon(n)$

TL;DR

This paper proves exponential decay of loop sizes in the loop $O(n)$ model on the hexagonal lattice for $n>1$ and certain $x$, indicating a phase transition at a critical $x_c(n)$ greater than $1/\sqrt{3}$ for $n o 1$ to 2.

Contribution

It establishes exponential decay for the loop $O(n)$ model in a new parameter regime, extending known results from the $n=1$ case to $n>1$ using FK-Ising properties.

Findings

Exponential decay of loop sizes for $n>1$ and $x<1/\sqrt{3}+ ext{small}$

Critical parameter $x_c(n)$ is strictly greater than $1/\sqrt{3}$ for $n o 1$ to 2

Proof leverages FK-Ising exponential decay and decomposing $n$ as $1+(n-1)$

Abstract

We show that the loop $O(n)$ model on the hexagonal lattice exhibits exponential decay of loop sizes whenever $n> 1$ and $x<\tfrac{1}{\sqrt{3}}+\varepsilon(n)$, for some suitable choice of $\varepsilon(n)>0$. It is expected that, for $n \leq 2$, the model exhibits a phase transition in terms of~$x$, that separates regimes of polynomial and exponential decay of loop sizes. In this paradigm, our result implies that the phase transition for $n \in (1,2]$ occurs at some critical parameter $x_c(n)$ strictly greater than that $x_c(1) = 1/\sqrt3$. The value of the latter is known since the loop $O(1)$ model on the hexagonal lattice represents the contours of the spin-clusters of the Ising model on the triangular lattice. The proof is based on developing $n$ as $1+(n-1)$ and exploiting the fact that, when $x<\tfrac{1}{\sqrt{3}}$, the Ising model exhibits exponential decay on any (possibly non simply-connected) domain. The latter follows from the positive association of the FK-Ising representation.