Percolation transition for random forests in $d\geq 3$

Abstract

The arboreal gas is the probability measure on (unrooted spanning) forests of a graph in which each forest is weighted by a factor β > 0 per edge. It arises as the q → 0 limit with p = βq of the q-state random cluster model. We prove that in dimensions d 3 the arboreal gas undergoes a percolation phase transition. This contrasts with the case of d = 2 where all trees are finite for all β > 0.The starting point for our analysis is an exact relationship between the arboreal gas and a fermionic non-linear sigma model with target space H 0|2 . This latter model can be thought of as the 0-state Potts model, with the arboreal gas being its random cluster representation. Unlike the q > 0 Potts models, the H 0|2 model has continuous symmetries. By combining a renormalisation group analysis with Ward identities we prove that this symmetry is spontaneously broken at low temperatures. In terms of the arboreal gas, this symmetry breaking translates into the existence of infinite trees in the thermodynamic limit. Our analysis also establishes massless free field correlations at low temperatures and the existence of a macroscopic tree on finite tori.

Efficient Algorithms for Approximating Quantum Partition Functions at Low Temperature

“…However, efficient algorithms may still exist. For example, the arboreal gas (a model of interacting symplectic fermions) possesses highly degenerate ground states [33], but nonetheless an efficient algorithm exists at all temperatures [34].…”

Forests on wired regular trees

“…The random forest model on complete graphs was studied in [6,8] and more recently in the regular lattice Z d in [3] (for d = 2) and [2] (for d ≥ 3). We refer to these papers for more references and history of the model.…”

“…We refer to these papers for more references and history of the model. Let us mention that both the papers [3,2] use a connection between the two point function in the Arboreal gas model and a non linear sigma model with hyperbolic target space. In contrast, our techniques are quite elementary and relies heavily on the regularity of the tree which leads to an explicit recursion.…”

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].…”

“…In particular, the H 0|2 -model can be represented by a purely Fermionic model with a Gaussian measure perturbed by a short range interaction. This allows to apply a rigorous renormalization group analysis in [BCH21]. It is not clear how to extend such an analysis to the H 2|2 -model, the main obstruction being the hyperbolic structure and the presence of long range interactions.…”