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 β> 0 per edge. This is called the arboreal gas model, and the special case when β= 1 is the uniform forest model. The arboreal gas can equivalently be defined to be Bernoulli bond percolation with parameter p= β/ (1 + β) conditioned to be acyclic, or as the limit q→ 0 with p= βq of the random cluster model. It is known that on the complete graph K_N with β= α/ N there is a phase transition similar to that of the Erdős–Rényi random graph: a giant tree percolates for α> 1 and all trees have bounded size for 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 Z² for any finite 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.