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

We show that the loop $O(n)$ model exhibits exponential decay of loop sizes whenever $ngeq 1$ and $x 0$. It is expected that, for $n leq 2$, the model exhibits a phase transition 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 $x_c(1) = 1/sqrt3$. value of the latter is known since the loop $O(1)$ model on the hexagonal lattice represents the contours of 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 $xu003ctfrac{1}{sqrt{3}}$, the Ising model exhibits exponential decay on any (possibly non simply-connected) domain. latter follows from the positive association of the FK-Ising representation.