Random Spanning Forests and Hyperbolic Symmetry
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Communications in
Mathematical Physics
Random Spanning Forests and Hyperbolic Symmetry Roland Bauerschmidt1 , Nicholas Crawford2 , Tyler Helmuth3 , Andrew Swan1 1 Statistical Laboratory, DPMMS, University of Cambridge, Cambridge, UK.
E-mail: [email protected]; [email protected]
2 Department of Mathematics, The Technion, Haifa, Israel. E-mail: [email protected] 3 Department of Mathematical Sciences, Durham University, Durham, UK.
E-mail: [email protected] Received: 13 December 2019 / Accepted: 22 September 2020 © The Author(s) 2020
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 K N with β = α/N there is a phase transition similar to that of the Erd˝os–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. 1. The Arboreal Gas and Uniform Forest Model 1.1. Definition and main results. Let G = (, E) be a finite (undirected) graph. A forest is a subgraph F = (, E ) that does not contain any cycles. We write F for the set of all forests. For β > 0 the arboreal gas (or weighted uniform forest model) is the measure on forests F defined by Pβ [F] ≡
1 |F| β , Zβ
Zβ ≡
β |F| ,
(1.1)
F∈F
where |F| denotes the number of edges in F. It is an elementary observation that the arboreal gas with parameter β is precisely Bernoulli bond percolation with parameter
R. Bauerschmidt, N. Crawford, T. Helmuth, A. Swan
pβ = β/(1 + β) conditioned to be acyclic: perc
P pβ
|F|
pβ (1 − pβ )|E|−|F| β |F| F | acyclic ≡ |F| = |F| = Pβ [F]. |E|−|F| Fβ F pβ (1 − pβ )
(1.2)
The arboreal gas model is also the limit, as q → 0 with p = βq, of the q-state random cluster model, see [40]. The particular case β = 1 is the uniform forest model mentioned in, e.g., [25,26,31,40]. 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
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