Phase transition in the spanning-hyperforest model on complete hypergraphs

Bedini, Andrea; Caracciolo, Sergio; Sportiello, Andrea

doi:10.1016/j.nuclphysb.2009.07.008

Cited by 9 publications

(16 citation statements)

References 44 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The following theorem shows that on the complete graph the arboreal gas undergoes a transition very similar to the percolation transition, i.e., the Erdős-Rényi graph. As mentioned in the introduction, this result has been obtained previously [8,34,36]. We have included a proof only to illustrate the utility of the H 0|2 representation.…”

Section: Phase Transition On the Complete Graphsupporting

confidence: 55%

“…In the infinite-volume limit, the arboreal gas is a singular conditioning of bond percolation, and hence the existence of a percolation transition as β varies is non-obvious. However, on the complete graph it is known that there is a phase transition, see [8,34,36]. To illustrate some of our methods we will give a new proof of the existence of a transition.…”

Section: The Arboreal Gas and Uniform Forest Modelmentioning

confidence: 99%

“…The arboreal gas has received attention under various names. An important reference for our work is [13], along with subsequent works by subsets of these authors and collaborators [7,8,[14][15][16]28]. These authors considered the connection of the arboreal gas with the antiferromagnetic S 0|2 model.…”

Section: Related Literaturementioning

confidence: 99%

“…As mentioned before, considerably stronger results are known for the arboreal gas on the complete graph. The first result in this direction concerned forests with a fixed number of edges [34], and later a fixed number of trees was considered [8]. Later in [36] the arboreal gas itself was considered, in the guise of the Erdős-Rényi graph conditioned to be acyclic.…”

Section: Related Literaturementioning

confidence: 99%

See 3 more Smart Citations

Random Spanning Forests and Hyperbolic Symmetry

Bauerschmidt

Crawford

Helmuth

et al. 2020

Commun. Math. Phys.

View full text Add to dashboard Cite

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.

show abstract

Section: Phase Transition On the Complete Graphsupporting

confidence: 55%

Section: The Arboreal Gas and Uniform Forest Modelmentioning

confidence: 99%

Section: Related Literaturementioning

confidence: 99%

Section: Related Literaturementioning

confidence: 99%

See 2 more Smart Citations

Random Spanning Forests and Hyperbolic Symmetry

Bauerschmidt

Crawford

Helmuth

et al. 2020

Commun. Math. Phys.

View full text Add to dashboard Cite

show abstract

“…In [20] the representation (1.1) was generalized to study spanning hyperforests in a hypergraph; and in [9,10] explicit results were obtained for spanning hyperforests in the complete hypergraph.…”

Section: Introductionmentioning

confidence: 99%

Spanning forests andOSP(N|2M) -invariantσ-models

Caracciolo¹,

Sokal²,

Sportiello³

2017

J. Phys. A: Math. Theor.

Self Cite

View full text Add to dashboard Cite

The present paper is part of our ongoing work on OSP (N |2M ) supersymmetric σ-models, their relation with the Potts model at q = 0 and spanning forests, and the rigorous analytic continuation of the partition function as an entire function of N − 2M , a feature first predicted by Parisi and Sourlas in the 1970's. Here we accomplish two main steps. First, we analyze in detail the role of the Ising variables that arise when the constraint in the OSP (1|2) model is solved, and we point out two situations in which the Ising and forest variables decouple. Second, we establish the analytic continuation for the OSP (N |2M ) model in some special cases: when the underlying graph is a forest, and for the Nienhuis action on a cubic graph. We also make progress in understanding the series-parallel graphs. ∞ k=0 c k x k is a formal power series with coefficients in R or Z, we can apply it to any f ∈ R[χ] + because f is nilpotent and the sum is therefore 3 If the coefficient ring R contains an element 1 2 -as it will in all the cases considered herethen χ 2 i = 0 is of course a consequence of the i = j case of χ i χ j + χ j χ i = 0. 4 In [21, Proposition A.9] it is proven that is at most n/2 + 2.

show abstract

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

Phase transition in the spanning-hyperforest model on complete hypergraphs

Cited by 9 publications

References 44 publications

Random Spanning Forests and Hyperbolic Symmetry

Random Spanning Forests and Hyperbolic Symmetry

Spanning forests andOSP(N|2M) -invariantσ-models

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

Contact Info

Product

Resources

About