We show that the loop O(n) model exhibits exponential decay of loop sizes whenever n\geq 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 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.
