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 $$\beta >0$$ β > 0 per edge. This is called the arboreal gas model, and the special case when $$\beta =1$$ β = 1 is the uniform forest model. The arboreal gas can equivalently be defined to be Bernoulli bond percolation with parameter $$p=\beta /(1+\beta )$$ p = β / ( 1 + β ) conditioned to be acyclic, or as the limit $$q\rightarrow 0$$ q → 0 with $$p=\beta q$$ p = β q of the random cluster model. It is known that on the complete graph $$K_{N}$$ K N with $$\beta =\alpha /N$$ β = α / N there is a phase transition similar to that of the Erdős–Rényi random graph: a giant tree percolates for $$\alpha > 1$$ α > 1 and all trees have bounded size for $$\alpha <1$$ α < 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 $${\mathbb {Z}}^2$$ Z 2 for any finite $$\beta >0$$ β > 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.