Abstract
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 $Z^d$, $d
3$, when $N N_>0$ and the inverse temperature $\beta$ is
large enough. This is a new result when $N>2$ and extends the classical result
of Fröhlich, 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 $N ın
N_>0$, 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.
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