Forests on wired regular trees

Ray, Gourab; Xiao, Ben

doi:10.30757/alea.v19-42

Cited by 5 publications

(2 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Finally, we mention that a detailed analysis of the infinite volume behaviour of the arboreal gas on regular trees with wired boundary conditions has been carried out [44,71]. This infinite volume behaviour is consistent with the finite volume behaviour of the complete graph, e.g., at all supercritical temperatures the sizes of finite clusters have the same distribution as those of critical percolation.…”

Section: Infinite Volume Behaviour and Relation To The Uniform Spanni...supporting

confidence: 70%

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

Bauerschmidt,

Crawford,

Helmuth

2024

Invent. math.

View full text Add to dashboard Cite

The arboreal gas is the probability measure on (unrooted spanning) forests of a graph in which each forest is weighted by a factor $\beta >0$ β > 0 per edge. It arises as the $q\to 0$ q → 0 limit of the $q$ q -state random cluster model with $p=\beta q$ p = β q . We prove that in dimensions $d\geqslant 3$ d ⩾ 3 the arboreal gas undergoes a percolation phase transition. This contrasts with the case of $d=2$ d = 2 where no percolation transition occurs.The starting point for our analysis is an exact relationship between the arboreal gas and a non-linear sigma model with target space the fermionic hyperbolic plane $\mathbb{H}^{0|2}$ 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 standard Potts models, the $\mathbb{H}^{0|2}$ 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.

show abstract

Section: Infinite Volume Behaviour and Relation To The Uniform Spanni...supporting

confidence: 70%

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

Bauerschmidt,

Crawford,

Helmuth

2024

Invent. math.

View full text Add to dashboard Cite

show abstract

“…The authors also establish strong quantitative control of the model, showing in particular that the finite-cluster two-point function continues to display critical-like behaviour in the supercritical regime. (Similar phenomena have also been shown to occur for the arboreal gas on the complete graph [33,38] and on regular trees with wired boundary conditions [17,41], where the analysis of the critical-like behaviour of finite/non-giant clusters is more complete. )…”

Section: Introductionsupporting

confidence: 56%

Uniqueness of the infinite tree in low-dimensional random forests

Halberstam¹,

Hutchcroft²

2023

Preprint

View full text Add to dashboard Cite

The arboreal gas is the random (unrooted) spanning forest of a graph in which each forest is sampled with probability proportional to β #edges for some β ≥ 0, which arises as the q → 0 limit of the Fortuin-Kastelyn random cluster model with p = βq. We study the infinite-volume limits of the arboreal gas on the hypercubic lattice Z d , and prove that when d ≤ 4, any translation-invariant infinite volume Gibbs measure contains at most one infinite tree almost surely. Together with the existence theorem of Bauerschmidt, Crawford and Helmuth (2021), this establishes that for d = 3, 4 there exists a value of β above which subsequential weak limits of the β-arboreal gas on tori have exactly one infinite tree almost surely. We also show that the infinite trees of any translation-invariant Gibbs measure on Z d are one-ended almost surely in every dimension. The proof has two main ingredients: First, we prove a resampling property for translation-invariant arboreal gas Gibbs measures in every dimension, stating that the restriction of the arboreal gas to the trace of the union of its infinite trees is distributed as the uniform spanning forest on this same trace. Second, we prove that the uniform spanning forest of any translation-invariant random connected subgraph of Z d is connected almost surely when d ≤ 4. This proof also provides strong heuristic evidence for the conjecture that the supercritical arboreal gas contains infinitely many infinite trees in dimensions d ≥ 5. Along the way, we give the first systematic and axiomatic treatment of Gibbs measures for models of this form including the random cluster model and the uniform spanning tree.

show abstract

The wired arboreal gas on regular trees

Easo

2022

Electron. Commun. Probab.

View full text Add to dashboard Cite

We study the weak limit of the arboreal gas along any exhaustion of a regular tree with wired boundary conditions. We prove that this limit exists, does not depend on the choice of exhaustion, and undergoes a phase transition. Below and at criticality, we prove the model is equivalent to bond percolation. Above criticality, we characterise the model as the superposition of critical bond percolation and a random collection of infinite one-ended paths. This provides a simple example of an arboreal gas model that continues to exhibit critical-like behaviour throughout its supercritical phase.

show abstract

Forests on wired regular trees

Cited by 5 publications

References 8 publications

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

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

Uniqueness of the infinite tree in low-dimensional random forests

The wired arboreal gas on regular trees

Contact Info

Product

Resources

About