The wired arboreal gas on regular trees

Easo, Philip

doi:10.1214/22-ecp460

Cited by 5 publications

(2 citation statements)

References 11 publications

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“…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: 72%

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

Bauerschmidt,

Crawford,

Helmuth

2024

Invent. math.

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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.

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

confidence: 72%

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

Bauerschmidt,

Crawford,

Helmuth

2024

Invent. math.

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“…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: 55%

Uniqueness of the infinite tree in low-dimensional random forests

Halberstam¹,

Hutchcroft²

2023

Preprint

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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.

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Loop-erased partitioning of a graph: mean-field analysis

Avena

Gaudillière

Milanesi

et al. 2022

Electron. J. Probab.

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Given a weighted finite graph G, we consider a random partition of its vertex set induced by a measure on spanning rooted forests on G. The latter is a generalized parametric version of the classical Uniform Spanning Tree measure which can be sampled using loop-erased random walks stopped at a random independent exponential time of parameter q > 0. The related random trees-identifying the blocks of the partition-tend to cluster nodes visited by the random walk on time scale 1/q.We explore the emerging macroscopic structure by analyzing two-point correlations, as a function of the tuning parameter q. To this aim, it is defined an interaction potential between pair of vertices, as the probability that they do not belong to the same block. This interaction potential can be seen as an affinity measure for "densely connected nodes" and capture well-separated regions in network models presenting non-homogeneous landscapes. In this spirit, we compute this potential and its scaling limits on a complete graph and on a non-homogeneous weighted version with community structure. For such geometries we show phase-transitions in the behavior of the random partition as a function of the tuning parameter and the edge weights. Moreover, as a corollary of our main results, we infer the right scaling of the parameters that give rise to the emergence of "giant" blocks.

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The wired arboreal gas on regular trees

Cited by 5 publications

References 11 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

Loop-erased partitioning of a graph: mean-field analysis

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