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

We provide a uniformly-positive point-wise lower bound for the two-point function 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 $\beta$ is large enough. This is a new result when $N>2$ and extends the classical result of Fr\"ohlich, Simon and Spencer (1976). Our bound follows from a new site-monotonicity property of the two-point function which is of independent interest 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 walks and loops, including the loop $O(N)$ model, random lattice permutations, the dimer model, the double dimer model, and the loop representation of the classical spin $O(N)$ model.