On Clusters of Brownian Loops in d Dimensions

Werner, Wendelin

doi:10.1007/978-3-030-60754-8_33

Cited by 5 publications

(4 citation statements)

References 30 publications

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“…In view of [1,4] or Theorem 4.1 in [19], this indicates that β = γ = 1 and δ = 2 likely hold for such d, i.e. these exponents expectedly attain their meanfield values, see also [40] for related results. Note that if a mean-field regime is to appear for sufficiently large values of ν, it can only happen for ν 4 by (1.38).…”

Section: Corollary 16 (ν 1)mentioning

confidence: 92%

“…the percolation of excursion sets of ϕ| G , was initiated in [25] and more recently reinvigorated in [30]. The corresponding cable system free field ϕ and its connections with Poissonian ensembles of (continuous) loops and bi-infinite Brownian trajectories on G have recently been studied in [10,11,26,27,36,40]. Among these links, those relating ϕ to the model of random interlacements, introduced in [32], stand out.…”

Section: Introductionmentioning

confidence: 99%

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Critical exponents for a percolation model on transient graphs

2022

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We consider the bond percolation problem on a transient weighted graph induced by the excursion sets of the Gaussian free field on the corresponding cable system. Owing to the continuity of this setup and the strong Markov property of the field on the one hand, and the links with potential theory for the associated diffusion on the other, we rigorously determine the behavior of various key quantities related to the (near-)critical regime for this model. In particular, our results apply in case the base graph is the three-dimensional cubic lattice. They unveil the values of the associated critical exponents, which are explicit but not mean-field and consistent with predictions from scaling theory below the upper-critical dimension.

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Section: Corollary 16 (ν 1)mentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

Critical exponents for a percolation model on transient graphs

2022

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“…In this article, we consider the Gaussian free field ϕ on the cable system G associated to an arbitrary transient weighted graph G; see the discussion around (1.1) below for the precise setup. Cable processes have increasingly proved an insightful object of study, as shown for instance in the recent articles [7,8,19,21,27] and [29]. In the present work, we investigate a well-chosen observable, the capacity of finite clusters in the excursion set E ≥h of ϕ above height h ∈ R, see (1.5) below.…”

Section: Introductionmentioning

confidence: 99%

Cluster capacity functionals and isomorphism theorems for Gaussian free fields

Drewitz

Prévost

Rodriguez

2021

Probab. Theory Relat. Fields

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We investigate level sets of the Gaussian free field on continuous transient metric graphs $$\widetilde{{\mathcal {G}}}$$ G ~ and study the capacity of its level set clusters. We prove, without any further assumption on the base graph $${\mathcal {G}}$$ G , that the capacity of sign clusters on $$\widetilde{{\mathcal {G}}}$$ G ~ is finite almost surely. This leads to a new and effective criterion to determine whether the sign clusters of the free field on $$\widetilde{{\mathcal {G}}}$$ G ~ are bounded or not. It also elucidates why the critical parameter for percolation of level sets on $$\widetilde{{\mathcal {G}}}$$ G ~ vanishes in most instances in the massless case and establishes the continuity of this phase transition in a wide range of cases, including all vertex-transitive graphs. When the sign clusters on $$\widetilde{{\mathcal {G}}}$$ G ~ do not percolate, we further determine by means of isomorphism theory the exact law of the capacity of compact clusters at any height. Specifically, we derive this law from an extension of Sznitman’s refinement of Lupu’s recent isomorphism theorem relating the free field and random interlacements, proved along the way, and which holds under the sole assumption that sign clusters on $$\widetilde{{\mathcal {G}}}$$ G ~ are bounded. Finally, we show that the law of the cluster capacity functionals obtained in this way actually characterizes the isomorphism theorem, i.e. the two are equivalent.

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“…The utility of identities like (1.20) for problems in statistical mechanics cannot be over-emphasized, where it can for instance be used as a powerful dictionary between the worlds of percolation and random walks in transient setups, see e.g. [55] for early works in this direction, and more recently [44,67,25,68,27,26] see also [56,5] for percolation and first-passage percolation in the context of ∇ϕ-models, and refs. at the beginning of this introduction for a host of other applications.…”

Section: Introductionmentioning

confidence: 99%

An isomorphism theorem for Ginzburg-Landau interface models and scaling limits

Deuschel¹,

Rodriguez²

2022

Preprint

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We introduce a natural measure on bi-infinite random walk trajectories evolving in a time-dependent environment driven by the Langevin dynamics associated to a gradient Gibbs measure with convex potential. We derive an identity relating the occupation times of the Poissonian cloud induced by this measure to the square of the corresponding gradient field, which -generically -is not Gaussian. In the quadratic case, we recover a well-known generalization of the second Ray-Knight theorem. We further determine the scaling limits of the various objects involved in dimension 3, which are seen to exhibit homogenization. In particular, we prove that the renormalized square of the gradient field converges under appropriate rescaling to the Wick-ordered square of a Gaussian free field on R 3 with suitable diffusion matrix, thus extending a celebrated result of Naddaf and Spencer regarding the scaling limit of the field itself.

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On Clusters of Brownian Loops in d Dimensions

Cited by 5 publications

References 30 publications

Critical exponents for a percolation model on transient graphs

Critical exponents for a percolation model on transient graphs

Cluster capacity functionals and isomorphism theorems for Gaussian free fields

An isomorphism theorem for Ginzburg-Landau interface models and scaling limits

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