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.
