Random spanning forests and hyperbolic symmetry

Bauerschmidt, Roland; Crawford, Nick; Helmuth, Tyler; Swan, Andrew

Abstract

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 KN 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.

Citation

Bauerschmidt, R., Crawford, N., Helmuth, T., & Swan, A. (2020). Random spanning forests and hyperbolic symmetry. Communications in Mathematical Physics, 381, 1223-1261. https://doi.org/10.1007/s00220-020-03921-y

• Journal Article Type: Article
• Acceptance Date: Sep 23, 2020
• Online Publication Date: Dec 2, 2020
• Publication Date: Dec 2, 2020
• Deposit Date: Sep 24, 2020
• Publicly Available Date: Feb 5, 2021
• Journal: Communications in Mathematical Physics
• Print ISSN: 0010-3616
• Electronic ISSN: 1432-0916
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 381
• Pages: 1223-1261
• DOI: https://doi.org/10.1007/s00220-020-03921-y
• Public URL: https://durham-repository.worktribe.com/output/1261722
• Related Public URLs: https://arxiv.org/abs/1912.04854