Random spanning forests and hyperbolic symmetry

We study (unrooted) random forests on a graph where the probability of a forest is multiplicatively weighted by a parameter $\beta>0$ per edge. This is called the arboreal gas model, and the special case when $\beta=1$ is the uniform forest model. The arboreal gas can equivalently be defined to be Bernoulli bond percolation with parameter $p=\beta/(1+\beta)$ conditioned to be acyclic, or as the limit $q\to 0$ with $p=\beta q$ of the random cluster model. It is known that on the complete graph $K_{N}$ with $\beta=\alpha/N$ there is a phase transition similar to that of the Erd\H{o}s--R\'enyi random graph: a giant tree percolates for $\alpha > 1$ and all trees have bounded size for $\alpha<1$. In contrast to this, by exploiting an exact relationship between the arboreal gas and a supersymmetric sigma model with hyperbolic target space, we show that the forest constraint is significant in two dimensions: trees do not percolate on $\mathbb{Z}^2$ for any finite $\beta>0$. This result is a consequence of a Mermin--Wagner theorem associated to the hyperbolic symmetry of the sigma model. Our proof makes use of two main ingredients: techniques previously developed for hyperbolic sigma models related to linearly reinforced random walks and a version of the principle of dimensional reduction.