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We provide a uniformly-positive point-wise lower bound for the two-pointfunction of the classical spin $O(N)$ model on the torus of $\mathbb{Z}^d$, $d\geq 3$, when $N \in \mathbb{N}_{>0}$ and the inverse temperature $\beta$ islarge enough. This is a new result when $N>2$ and extends the classical resultof Fr\"ohlich, Simon and Spencer (1976). Our bound follows from a newsite-monotonicity property of the two-point function which is of independentinterest and holds not only for the spin $O(N)$ model with arbitrary $N \in\mathbb{N}_{>0}$, but for a wide class of systems of interacting random walksand loops, including the loop $O(N)$ model, random lattice permutations, thedimer model, the double dimer model, and the loop representation of theclassical spin $O(N)$ model.

Site monotonicity and uniform positivity for interacting random walks and the spin O(N) model with arbitrary N