Percolation for 2D Classical Heisenberg Model and Exit Sets of Vector Valued GFF

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 Gaussian Free Field: $\phi : \mathbb{Z}^2 \rightarrow \mathbb{R}^N$, $N \ge 1$. We prove that, somewhat surprisingly, these sets are a.s. degenerate as long as $N \ge 2$, while they are conjectured to be macroscopic and fractal when $N = 1$. This analysis allows us, when $N \ge 2$, to understand the percolation properties of the level sets of ${ \Vert \phi(x) \Vert_2, x \in \mathbb{Z}^2 }$ and leads us to our second main motivation in this work: if one projects a spin $O(N+1)$ model (the case $N = 2$ corresponds to the classical Heisenberg model) down to a spin $O(N)$ model, we end up with a spin $O(N)$ in a quenched disorder given by random conductances on $\mathbb{Z}^2$. Using the exit sets of the $N$-vector-valued GFF, we obtain a local and geometric description of this random disorder in the limit $\beta \rightarrow \infty$. This allows us in particular to revisit a series of celebrated works by Patrascioiu and Seiler (J Stat Phys 69(3):573–595, 1992; Nucl Phys B Proc Suppl 30:184–191, 1993; J Stat Phys 106(3):811–826, 2002) which argued against Polyakov's prediction that the spin $O(N+1)$ model is massive at all temperatures as long as $N \ge 2$ (Polyakov in Phys Lett B 59(1):79–81, 1975). 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 fluctuations of a classical Heisenberg model near a north-pointing spin are given by a $N = 2$ vectorial GFF. This is implicit, for example, in Polyakov (1975), but we give here the first (non-trivial) rigorous proof. Also, independently of the recent work Dubédat and Falconet (Random clusters in the Villain and XY models, arXiv preprint arXiv:2210.03620, 2022), we show that two-point correlation functions of the spin $O(N)$ model can be given in terms of certain percolation events in the cable graph for any $N \ge 1$.