On the radius of Gaussian free field excursion clusters

Goswami, Subhajit; Rodriguez, Pierre-François; Severo, Franco

doi:10.1214/22-aop1569

Cited by 16 publications

(6 citation statements)

References 37 publications

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“…The methods we develop here are robust, and as such, provide a template that paves the way toward a better understanding of the supercritical regime in other dependent models of interest, including 'non-elliptic' ones, for example, violating finite-energy property; matters relating to the 'well-behavedness' of the supercritical phase have so far witnessed comparatively little progress, and results are restricted to specific models [5,8,14,15,20,22,32] and references therein. To wit, the gluing technique we develop here yields a more robust proof of [15,Proposition 4.1], which avoids the use of a very specific decomposition of the field, and relies overall on a much less precise understanding of the conditional behavior of the occupation field.…”

Section: Proof Outlinementioning

confidence: 99%

“…= 𝑣 ′ −𝑣 3 ∧ 𝑢−𝑣 ′ 3 . We will apply the results of [10] to the graph ℤ 𝑑 (with unit weights); see also the proof of [20,Lemma 5.16] for a similar argument in the context of the Gaussian free field. Let 𝐿 0 = 𝑀(𝑁 0 ) for a large positive integer parameter 𝑁 0 to be determined.…”

Section: Strong Percolation From Gluing Propertymentioning

confidence: 99%

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A characterization of strong percolation via disconnection

Duminil‐Copin,

Goswami,

Rodriguez

et al. 2024

Proceedings of London Math Soc

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We consider a percolation model, the vacant set of random interlacements on , , in the regime of parameters in which it is strongly percolative. By definition, such values of pinpoint a robust subset of the supercritical phase, with strong quantitative controls on large local clusters. In the present work, we give a new characterization of this regime in terms of a single property, monotone in , involving a disconnection estimate for . A key aspect is to exhibit a gluing property for large local clusters from this information alone, and a major challenge in this undertaking is the fact that the conditional law of exhibits degeneracies. As one of the main novelties of this work, the gluing technique we develop to merge large clusters accounts for such effects. In particular, our methods do not rely on the widely assumed finite‐energy property, which the set does not possess. The characterization we derive plays a decisive role in the proof of a lasting conjecture regarding the coincidence of various critical parameters naturally associated to in the companion article [Duminil‐Copin, Goswami, Rodriguez, Severo, and Teixeira, Phase transition for the vacant set of random walk and random interlacements, arXiv:2308.07919, 2023].

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Section: Proof Outlinementioning

confidence: 99%

Section: Strong Percolation From Gluing Propertymentioning

confidence: 99%

A characterization of strong percolation via disconnection

Duminil‐Copin,

Goswami,

Rodriguez

et al. 2024

Proceedings of London Math Soc

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“…We now return to the lower bound(s) in (1.27) (and (1.46) below) and their proofs, which are instructive. In both cases, we rely on a change-of-measure argument, somewhat similar to the one used in [15], but quantitative (the arguments in [15] operate at fixed level a as r → ∞); see also [5,35], for arguments of this kind in various contexts involving ϕ. We modify P so as to shift a given level a ∈ (0, 1] to −a, which is (slightly) supercritical, in an appropriate region.…”

Section: Corollary 15 (Scaling Relation)mentioning

confidence: 99%

“…An essential tool in obtaining lower bounds on various probabilities for the Gaussian free field has been a certain change-of-measure formula, see [5,15,35] or (6.17) for instance. In this appendix, we present another version of this formula when studying events only depending on the cluster K a from (1.4), which is useful in the proof of Lemma 6.2.…”

Section: A Appendix: An Enhanced Change-of-measure Formulamentioning

confidence: 99%

“…We refer to ( Regarding lower bounds for related quantities ψ, τ tr a , cf. (1.45), which include the regime ν > 1, we refer to the discussion around Theorem 1.7 at the very end of this introduction and to the recent article [15] concerning results related to (1.26) and (1.27) for the discrete problem on Z 3 , which witness ξ = ξ(a) in yielding the bounds cξ(a) −1 − log(r ) r log ψ(a, r ) c ξ(a) −1 valid for all large enough r R(a) but lack any quantitative control on R(a). The version of (1.26) for τ tr a including the sharp pre-factor as a → 0 will be crucial for our purposes, see Corollary 1.5 below.…”

Section: Introductionmentioning

confidence: 99%

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Critical exponents for a percolation model on transient graphs

2022

Self Cite

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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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Disconnection and Entropic Repulsion for the Harmonic Crystal with Random Conductances

Chiarini

Nitzschner

2021

Commun. Math. Phys.

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We study level-set percolation for the harmonic crystal on Z d , d ≥ 3, with uniformly elliptic random conductances. We prove that this model undergoes a nontrivial phase transition at a critical level that is almost surely constant under the environment measure. Moreover, we study the disconnection event that the level-set of this field below a level α disconnects the discrete blow-up of a compact set A ⊆ R d from the boundary of an enclosing box. We obtain quenched asymptotic upper and lower bounds on its probability in terms of the homogenized capacity of A, utilizing results from Neukamm, Schäffner and Schlömerkemper (SIAM J Math Anal 49(3):1761-1809, 2017). Furthermore, we give upper bounds on the probability that a local average of the field deviates from some profile function depending on A, when disconnection occurs. The upper and lower bounds concerning disconnection that we derive are plausibly matching at leading order. In this case, this work shows that conditioning on disconnection leads to an entropic push-down of the field. The results in this article generalize the findings of Nitzschner (Electron J Probab 23:105, 2018) and Chiarini and Nitzschner (Probab Theory Relat Fields 177(1-2):525-575, 2020) which treat the case of constant conductances. Our proofs involve novel "solidification estimates" for random walks, which are similar in nature to the corresponding estimates for Brownian motion derived by Nitzschner and Sznitman (

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On the radius of Gaussian free field excursion clusters

Cited by 16 publications

References 37 publications

A characterization of strong percolation via disconnection

A characterization of strong percolation via disconnection

Critical exponents for a percolation model on transient graphs

Disconnection and Entropic Repulsion for the Harmonic Crystal with Random Conductances

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