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We show that the loop $O(n)$ model on the hexagonal lattice exhibitsexponential 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 interms of~$x$, that separates regimes of polynomial and exponential decay ofloop 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 thanthat $x_c(1) = 1/\sqrt3$. The value of the latter is known since the loop$O(1)$ model on the hexagonal lattice represents the contours of thespin-clusters of the Ising model on the triangular lattice.  The proof is based on developing $n$ as $1+(n-1)$ and exploiting the factthat, when $x<\tfrac{1}{\sqrt{3}}$, the Ising model exhibits exponential decayon any (possibly non simply-connected) domain. The latter follows from thepositive association of the FK-Ising representation.

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