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 $\beta >0$ 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 $q\to 0$ 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.

arXiv commentAccepted version

References in corpus (13)

• Riemannian symmetric superspaces and their origin in random matrix theory
• The Random-Cluster Model
• Quasi-diffusion in a 3D Supersymmetric Hyperbolic Sigma Model
• Localization for Linearly Edge Reinforced Random Walks
• The arboreal gas and the supersphere sigma model
• Ferromagnetic phase transition for the spanning-forest model (q \to 0 limit of the Potts model) in three or more dimensions
• Anderson localization for a supersymmetric sigma model
• Mean field frozen percolation
• Spontaneous symmetry breaking of a hyperbolic sigma model in three dimensions
• Grassmann Integral Representation for Spanning Hyperforests
• Dynkin isomorphism and Mermin--Wagner theorems for hyperbolic sigma models and recurrence of the two-dimensional vertex-reinforced jump process
• Phase transition in the spanning-hyperforest model on complete hypergraphs
• Hyperforests on the Complete Hypergraph by Grassmann Integral Representation

Cited by in corpus (7)

• Ginzburg-Landau Description and Emergent Supersymmetry of the $(3,8)$ Minimal Model
• Critical Field Theories with OSp$(1|2M)$ Symmetry
• Sphere free energy of scalar field theories with cubic interactions
• Uniqueness of the infinite tree in low-dimensional random forests
• Percolation transition for random forests in $d\ge 3$
• Dualities between fermionic theories and the Potts model
• Anomalous criticality coexists with giant cluster in the uniform forest model