  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.
