Shifted critical threshold in the loop O(n) model at arbitrary small n

In the loop $O(n)$ model a collection of mutually-disjoint self-avoiding loops is drawn at random on a finite domain of a lattice with probability proportional to $${\lambda^{\# \mbox{edges}} n^{\# \mbox{loops}},}$$ where $\lambda, n \in [0, \infty)$. Let $\mu$ be the connective constant of the lattice and, for any $n \in [0, \infty)$, let $\lambda_c(n)$ be the largest value of $\lambda$ such that the loop length admits uniformly bounded exponential moments. It is not difficult to prove that $\lambda_c(n) =1/\mu$ when $n=0$ (in this case the model corresponds to the self-avoiding walk) and that for any $n \geq 0$, $\lambda_c(n) \geq 1/\mu$. In this note we prove that, \begin{align*} \lambda_c(n) & > 1/\mu \, \, \, \, \, \, \, \, \, \, \, \mbox{whenever $n >0$}, \\ \lambda_c(n) & \geq 1/\mu \, + \, c_0 \, n \, + \, O(n^2), \end{align*} on $\mathbb{Z}^d$, with $d \geq 2$, and on the hexagonal lattice, where $c_0>0$. This means that, when $n$ is positive (even arbitrarily small), as a consequence of the mutual repulsion between the loops, a phase transition can only occur at a strictly larger critical threshold than in the self-avoiding walk.