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 ℤ^푑, 푑≥3, when 푁∈ℕ>0 and the inverse temperature 훽 is large enough. This is a new result when 푁>2 and extends the classical result of Fröhlich et al. (Commun Math Phys 50:79–95, 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 푁∈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.