Exponential Decay in the Loop O(n) Model on the Hexagonal Lattice for n > 1 and $$x<\tfrac {1}{\sqrt {3}}+\varepsilon (n)$$

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 ε(n) > 0.It is expected that, for n ≤ 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 ∈ (1, 2] occurs at some critical parameter xc(n) strictly greater than that \(x_c(1) = 1/\sqrt {3}\). 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.