Percolation for 2D classical Heisenberg model and exit sets of vector valued GFF

Abstract

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 $N\geq 1$.