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Random spanning forests and hyperbolic symmetry

Bauerschmidt, Roland and Crawford, Nick and Helmuth, Tyler and Swan, Andrew (2020) 'Random spanning forests and hyperbolic symmetry.', Communications in mathematical physics., 381 . pp. 1223-1261.


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.

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Publisher statement:Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit
Date accepted:23 September 2020
Date deposited:24 September 2020
Date of first online publication:02 December 2020
Date first made open access:05 February 2021

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