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 $${\lambda }^{\mathrm{\#}edges}{n}^{\mathrm{\#}loops},$$ where $\lambda ,n\in [0,\mathrm{\infty })$. Let $\mu$ be the connective constant of the lattice and, for any $n\in [0,\mathrm{\infty })$, 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 ${\lambda }_c(n)=1\mathrm{/}\mu$ when $n=0$ (in this case the model corresponds to the self-avoiding walk) and that for any $n\ge 0$, ${\lambda }_c(n)\ge 1\mathrm{/}\mu$. 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 $d\ge 2$, 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.