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 $$\beta >0$$ β > 0 per edge. This is called the arboreal gas model, and the special case when $$\beta =1$$ β = 1 is the uniform forest model. The arboreal gas can equivalently be defined to be Bernoulli bond percolation with parameter $$p=\beta /(1+\beta )$$ p = β / ( 1 + β ) conditioned to be acyclic, or as the limit $$q\rightarrow 0$$ q → 0 with $$p=\beta q$$ p = β q of the random cluster model. It is known that on the complete graph $$K_{N}$$ K N with $$\beta =\alpha /N$$ β = α / N there is a phase transition similar to that of the Erdős–Rényi random graph: a giant tree percolates for $$\alpha > 1$$ α > 1 and all trees have bounded size for $$\alpha <1$$ 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 $${\mathbb {Z}}^2$$ Z 2 for any finite $$\beta >0$$ 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.

Spin systems with hyperbolic symmetry: a survey

“…The H 2 and H 2|2 models, motivated by the Anderson transition [34,70], are exactly related to (linearly) edge-reinforced random walks and vertex-reinforced jump processes, introduced independently in the probability literature in the 1980s [29] and early 2000s [26]. A similar connection exists between the related H 0|2 model and random forests [11,22]; random forests arose earlier (for example) in connection with the Fortuin-Kasteleyn random cluster model [43]. The connections between hyperbolic spin systems and probabilistic phenomena are the main topic of this survey.…”

“…There is a natural generalisation of the above models to the broader class of H n|2m models with n + 1 commuting coordinates and 2m anticommuting coordinates. Generalising Theorem 2.1, there is an exact correspondence between observables of the H n|2m and H n+2|2m+2 models, see [11,Section 2]. For developments when n = 0, see [25].…”

“…Here x i and (ξ i , η i ) stand for an even (bosonic) coordinate and pair of odd (fermionic) coordinates when n, m > 1, respectively. The proofs of these identities boil down to integration by parts, see e.g., [34,Appendix B] or [11,Lemma 2.3].…”

“…For the H 2 model, SSB occurs for any β > 0 as a result of convexity [48]. The H 2|2 and H 0|2 models, however, have phase transitions: SSB in the form of (2.15) and (2.16), respectively, occurs in d ≥ 3 if and only if β is sufficiently large [10,34]. Once SSB is known to occur (or not), it is interesting and physically relevant to ask more precise questions, e.g., about the asymptotics of the correlation functions x i x j β,h .…”

The non-linear supersymmetric hyperbolic sigma model on a complete graph with hierarchical interactions

“…For p = 2, these models are examined first by Crawford in [Cra21]. The case of the H 0|2 -model, which describes the arboreal gas (configurations of unrooted spanning forests), and the closely related H 2|4 -model are examined by Bauerschmidt, Crawford, Helmuth, and Swan in [BCHS21] and [BCH21].…”