Random Spanning Forests and Hyperbolic Symmetry

Abstract

We study (unrooted) random forests on a graph where the probability of a forest is multiplicatively weighted by a parameter β>0$$\beta >0$$ per edge. This is called the arboreal gas model, and the special case when β=1$$\beta =1$$ is the uniform forest model. The arboreal gas can equivalently be defined to be Bernoulli bond percolation with parameter p=β/(1+β)$$p=\beta /(1+\beta )$$ conditioned to be acyclic, or as the limit q→0$$q\rightarrow 0$$ with p=βq$$p=\beta q$$ of the random cluster model. It is known that on the complete graph KN$$K_{N}$$ with β=α/N$$\beta =\alpha /N$$ there is a phase transition similar to that of the Erdős–Rényi random graph: a giant tree percolates for α>1$$\alpha > 1$$ and all trees have bounded size for α<1$$\alpha <1$$. 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 Z2$${\mathbb {Z}}^2$$ for any finite β>0$$\beta >0$$. 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.