  The arboreal gas is the probability measure on (unrooted spanning) forests of
a graph in which each forest is weighted by a factor $\beta>0$ per edge. It
arises as the $q\to 0$ limit of the $q$-state random cluster model with
$p=\beta q$. We prove that in dimensions $d\geq 3$ the arboreal gas undergoes a
percolation phase transition. This contrasts with the case of $d=2$ where no
percolation transition occurs.
  The starting point for our analysis is an exact relationship between the
arboreal gas and a non-linear sigma model with target space the fermionic
hyperbolic plane $\mathbb{H}^{0|2}$. This latter model can be thought of as the
$0$-state Potts model, with the arboreal gas being its random cluster
representation. Unlike the standard Potts models, the $\mathbb{H}^{0|2}$ model
has continuous symmetries. By combining a renormalisation group analysis with
Ward identities we prove that this symmetry is spontaneously broken at low
temperatures. In terms of the arboreal gas, this symmetry breaking translates
into the existence of infinite trees in the thermodynamic limit. Our analysis
also establishes massless free field correlations at low temperatures and the
existence of a macroscopic tree on finite tori.
