  Our motivation in this paper is twofold. First, we study the geometry of a
class of exploration sets, called exit sets, which are naturally associated
with a 2D vector-valued GFF : $\phi : Z^2 \to R^N, N\geq 1$. We prove that,
somewhat surprisingly, these sets are a.s. degenerate as long as $N\geq 2$,
while they are conjectured to be macroscopic and fractal when $N=1$.
  This analysis allows us, when $N\geq 2$, to understand the percolation
properties of the level sets of $\{\|\phi(x)\|, x\in Z^2\}$ and leads us to our
second main motivation in this work: if one projects a spin $O(N+1)$ model
(classical Heisenberg model is $N=2$) down to a spin $O(N)$ model, we end up
with a spin $O(N)$ in a quenched disorder given by random conductances on
$Z^2$. Using the exit sets of the $N$-vector-valued GFF, we obtain a local and
geometric description of this random disorder in the limit $\beta\to \infty$.
This allows us to revisit a series of celebrated works by Patrascioiu and
Seiler ([PS92, PS93, PS02]) which argued against Polyakov's prediction that
spin $O(N+1)$ model is massive at all temperatures when $N\geq 2$ ([Pol75]). We
make part of their arguments rigorous and more importantly we provide the
following counter-example: we build ergodic environments of (arbitrary) high
conductances with (arbitrary) small and disconnected regions of low
conductances in which, despite the predominance of high conductances, the $XY$
model remains massive.
  Of independent interest, we prove that at high $\beta$, the transverse
fluctuations of a classical Heisenberg model are given by a $N=2$ vectorial
GFF. This is implicit in [Pol75] but we give here the first (non-trivial)
rigorous proof. Also, independently of the recent work [DF22], we show that
two-point correlation functions of the spin $O(N)$ model are given in terms of
certain percolation events in the cable graph for any $N\geq 1$.
