Level-set percolation for the Gaussian free field on a transient tree

Abächerli, Angelo

doi:10.1214/16-aihp799

Cited by 26 publications

(45 citation statements)

References 14 publications

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“…Finally, in section 3, we extend the result of Bricmont, Lebowitz and Maes [5] to transient simple symmetric random walks on any regular graph. This extension has been already noticed by Abächerli and Sznitman (Proposition A2 in [1]). In [9], Drewitz, Prévost and Rodriguez go further by showing that h * > 0 for a large class of graphs.…”

Section: Introductionsupporting

confidence: 75%

Some percolations involving the Gaussian free fields

Eisenbaum

2021

Electron. Commun. Probab.

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Consider an infinite, connected, locally finite graph with vertex set V. Intuitively a simple point process on V with attractive properties, should percolate more easily than a Bernoulli point process with the same marginales. Although it seems wrong to imagine that it could be true in general, we confirm this intuition on several examples involving Gaussian free fields and permanental free fields.

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Section: Introductionsupporting

confidence: 75%

Some percolations involving the Gaussian free fields

Eisenbaum

2021

Electron. Commun. Probab.

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“…x is the effective resistance between x and infinity for the descendants of x, see Proposition 2.2 in [1]. These graphs verify (Cap) by Lemma 3.4,3).…”

Section: Introductionsupporting

confidence: 65%

“…Extending the setting in which the identity (Isom) is valid is also interesting as this relation has already been useful in [27] and [1] to compare the critical parameter for the percolation of random interlacements and the Gaussian free field on discrete trees, and in [8] to prove strong percolation for the level sets of the discrete Gaussian free field at a positive level on a large class graphs, for instance Z d , d ≥ 3, or various fractal graphs. It is not always easy to check that the conditions (1.32) and (1.34), or (1.42), of Theorem 2.4 in [27] are exactly verified, see the proof of Corollary 5.3 in [8] which sparked our interest, and it can thus be interesting to replace them by the weaker condition (Cap), which is easier to verify.…”

Section: (B) (A)mentioning

confidence: 99%

“…An interesting and immediate consequence of Lemma 2.3 and (2.22) is the following: if K n , n ∈ N, and K are compacts of G such that (K n ) n∈N increases or decreases to K , consider the quantity x∈∂ K e K (x)η 1 K n (x) in case the K n are increasing and x∈∂ K e K 1 (x)η 1 K n (x) in case the K n are decreasing, respectively (which both equal cap(K n ) by virtue of (2.22)). We can then take n → ∞ while applying (2.25) with f = 1 to obtain that…”

Section: Lemma 23mentioning

confidence: 99%

“…The proof of Theorem 3.2 appears over the next two sections. Note that the fact that E ≥h (x 0 ) is non-compact with positive probability for all h < 0 could alternatively be obtained from the Markov property (2.34) similarly as in [6], see also the Appendix of [1] for details, or from the isomorphism (1.7), see (1.8) and above. Here, we will obtain it as a direct consequence of our methods.…”

Section: Theorem 32mentioning

confidence: 99%

See 2 more Smart Citations

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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Giant component for the supercritical level‐set percolation of the Gaussian free field on regular expander graphs

Černý

2023

Comm Pure Appl Math

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We consider the zero‐average Gaussian free field on a certain class of finite d‐regular graphs for fixed . This class includes d‐regular expanders of large girth and typical realisations of random d‐regular graphs. We show that the level set of the zero‐average Gaussian free field above level h has a giant component in the whole supercritical phase, that is for all , with probability tending to one as the size of the graphs tends to infinity. In addition, we show that this component is unique. This significantly improves the result of [4], where it was shown that a linear fraction of vertices is in mesoscopic components if , and together with the description of the subcritical phase from [4] establishes a fully‐fledged percolation phase transition for the model.

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Level-set percolation for the Gaussian free field on a transient tree

Cited by 26 publications

References 14 publications

Some percolations involving the Gaussian free fields

Some percolations involving the Gaussian free fields

Cluster capacity functionals and isomorphism theorems for Gaussian free fields

Giant component for the supercritical level‐set percolation of the Gaussian free field on regular expander graphs

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