Spin systems with hyperbolic symmetry: a survey

Bauerschmidt, Roland; Helmuth, Tyler

doi:10.48550/arxiv.2109.02566

Cited by 4 publications

(6 citation statements)

References 63 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…2. If H is a finite connected subgraph of Z d then Q β (all vertices of H belong to the same augmented connectivity class | G H ) > 0 a.s. (6) We refer to the property (5) of Q β as deletion tolerance and the property (6) as merge tolerance.…”

Section: φ(H)mentioning

confidence: 99%

“…The deletion tolerance property ( 5) is an immediate consequence of the augmented Gibbs property since β < ∞. We now turn to the merge tolerance property (6). Since H is connected, H/Φ(H) is connected and therefore admits at least one spanning tree, which is given positive mass by the conditional measure P…”

Section: φ(H)mentioning

confidence: 99%

“…Despite these connections, there are very few tools available to study the model and several very basic conjectures about its behaviour have remained open for twenty years [23]. See [6,44] for surveys of the model and its connections to other topics. Interest in the arboreal gas has grown significantly in recent years following the breakthrough works of Bauerschmidt, Crawford, Helmuth and Swan [5] and Bauerschmidt, Crawford and Helmuth [4], who studied the model's percolation phase transition through the lens of spontaneous symmetry breaking in an equivalent supersymmetric hyperbolic sigma model: In [5] they proved that the arboreal gas on Z 2 never contains any infinite trees for any finite β < ∞, while in [4] they proved that the arboreal gas on Z d contains infinite trees for sufficiently large values of β when d ≥ 3.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Section: φ(H)mentioning

confidence: 99%

Section: φ(H)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Uniqueness of the infinite tree in low-dimensional random forests

Halberstam¹,

Hutchcroft²

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…Limits of the RCM giving rise to measures which are supported on forests are usually referred to as arboreal gases. Most of the literature on arboreal gas models is focused on unrooted forests; see, e.g., [11,12,13,34,18] for some very recent mathematical work in this direction. In [12, Appendix A] the authors consider what they call arboreal gas with an external field and they notice that this can be interpreted as a marginal of a measure over rooted spanning forests.…”

Section: Relations With the Random-cluster Modelmentioning

confidence: 99%

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

Avena

Gaudillière

Milanesi

et al. 2022

Electron. J. Probab.

View full text Add to dashboard Cite

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.

show abstract

“…Example 1.5 (Supersymmetric spin models). The Green kernel g ω (x, y) of an RCM of the form (1.7) with random killing appears in a respresentation for the two-point function of the supersymmetric hyperbolic sigma model, or H 2|2 -model, see [15] for a recent survey. The H 2|2 -model is a spin model introduced in [66] as a more tractable model for the Anderson transition.…”

mentioning

confidence: 99%

First passage percolation with long-range correlations and applications to random Schrödinger operators

Andrés¹,

Prévost²

2021

Preprint

View full text Add to dashboard Cite

We consider first passage percolation (FPP) with passage times generated by a general class of models with long-range correlations on Z d , d ≥ 2, including discrete Gaussian free fields, Ginzburg-Landau ∇φ interface models or random interlacements as prominent examples. We show that the associated time constant is positive, the FPP distance is comparable to the Euclidean distance, and we obtain a shape theorem. We also present two applications for random conductance models (RCM) with possibly unbounded and strongly correlated conductances. Namely, we obtain a Gaussian heat kernel upper bound for RCMs with a general class of speed measures, and an exponential decay estimate for the Green function of RCMs with random killing measures.

show abstract

Spin systems with hyperbolic symmetry: a survey

Cited by 4 publications

References 63 publications

Uniqueness of the infinite tree in low-dimensional random forests

Uniqueness of the infinite tree in low-dimensional random forests

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

First passage percolation with long-range correlations and applications to random Schrödinger operators

Contact Info

Product

Resources

About