Abstract
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 qrightarrow 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 Erdős–Rényi 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.
Highlights
Where |F| denotes the number of edges in F
We emphasize that the uniform forest model is not the weak limit of a uniformly chosen spanning tree; emphasis is needed since the latter model is called the ‘uniform spanning forest’ (USF) in the probability literature
We will shortly see that the arboreal gas has a richer phenomenology than the USF
Summary
In finite volume, the uniform spanning tree is the β → ∞ limit of the arboreal gas. Where β(t) is the graph Laplacian with edge weights βeti +t j , understood as acting on \ 0 This formula is a consequence of the hyperbolic sigma model representation of the arboreal gas. The following conjecture asserts that the arboreal gas has a phase transition in dimensions d 3, just as in mean-field theory (Proposition 1.2). Theorem 1.3 shows that the behaviour of the arboreal gas in two dimensions is different from that of percolation This difference would be considerably strengthened by the following conjecture, which first appeared in [13]. An important consequence of the full conjecture (with factor 1) is the existence of translation invariant arboreal gas measures on Zd ; we prove this in Appendix A. For the convenience of readers, we briefly discuss the fermionic representation of rooted spanning forests and spanning trees in Appendix B
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