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 $qto 0$ 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 ErdH{o}s--Renyi 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.
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