On coupling and “vacant set level set” percolation

Sznitman, Alain-Sol

doi:10.1214/19-ecp217

Cited by 6 publications

(4 citation statements)

References 16 publications

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“…Note that when r = 0, the situation for the Gaussian free field is different, see Remark 7.7, 2) and [9]. As explained in Section 1.1, the inequalities (2.27) were already proved on certain transient graphs for random interlacements and the Gaussian free field: h d * = 0 is proved in [18,41,48,60] on all vertex-transitive transient graphs, and in particular on Z d , d 3, or for the the pinned dGFF in dimension 2 ; h d * > 0 is proved in [1,2,16,17,62] on Z d , d 3, a class of fractal or Cayley graphs with polynomial growth, a large class of trees, and expander graphs; [16,55,56,63] on Z d , d 3, the same class of fractal or Cayley graphs with polynomial growth as before, and non-amenable graphs. However, the question of the strict inequality…”

Section: Discrete Resultsmentioning

confidence: 84%

“…On transient graphs, level set percolation for the dGFF has also received significant attention in recent years [51,60]. The isomorphism theorem relating its law with random interlacements has been a powerful tool for the study of the percolation of its level sets [16,17,61], and also implies the weak inequality h * √ 2u * between the respective critical parameters of the level sets of the dGFF and the vacant set of random interlacements, see [41,Theorem 3] on Z d , d 3, or [62,Corollary 2.5] and [18] on more general transient graphs. This inequality is strict on a large class of trees [1,60], and conjectured [60,Remark 4.6] to be strict on Z d for all d 3.…”

Section: Relation To Other Models and Further Motivationmentioning

confidence: 99%

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Percolation for two-dimensional excursion clouds and the discrete Gaussian free field

Drewitz¹,

Elias²,

Prévost³

et al. 2022

Preprint

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We study percolative properties of excursion processes and the discrete Gaussian free field (dGFF) in the planar unit disk. We consider discrete excursion clouds, defined using random walks as a two-dimensional version of random interlacements, as well as its scaling limit, defined using Brownian motion. We prove that the critical parameters associated to vacant set percolation for the two models are the same and equal to π/3. The value is obtained from a Schramm-Loewner evolution (SLE) computation. Via an isomorphism theorem, we use a generalization of the discrete result that also involves a loop soup (and an SLE computation) to show that the critical parameter associated to level set percolation for the dGFF is strictly positive and smaller than π/2. In particular this entails a strict inequality of the type h * < √ 2u * between the critical percolation parameters of the dGFF and the two-dimensional excursion cloud. Similar strict inequalities are conjectured to hold in a general transient setup.

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Section: Discrete Resultsmentioning

confidence: 84%

Section: Relation To Other Models and Further Motivationmentioning

confidence: 99%

Percolation for two-dimensional excursion clouds and the discrete Gaussian free field

Drewitz¹,

Elias²,

Prévost³

et al. 2022

Preprint

View full text Add to dashboard Cite

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“…More recent developments can be found for instance in [RS13], [PR15], [Szn15], [DPR18b], [DPR18a], [Nit18] and [CN18]. The particular case of Gaussian free field on regular trees was studied before in [Szn16] and [Szn19], and on general transient trees in [AS18]. Compared to the present article, the emphasis in these three papers is put on a different aspect of the Gaussian free field, namely its connection with the model of random interlacements.…”

Section: Introductionmentioning

confidence: 95%

Level-set percolation of the Gaussian free field on regular graphs I: Regular trees

Abächerli¹,

Černý²

2019

Preprint

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We study level-set percolation of the Gaussian free field on the infinite d-regular tree for fixed d ≥ 3. Denoting by h the critical value, we obtain the following results: for h > h we derive estimates on conditional exponential moments of the size of a fixed connected component of the level set above level h; for h < h we prove that the number of vertices connected over distance k above level h to a fixed vertex grows exponentially in k with positive probability. Furthermore, we show that the percolation probability is a continuous function of the level h, at least away from the critical value h . Along the way we also obtain matching upper and lower bounds on the eigenfunctions involved in the spectral characterisation of the critical value h and link the probability of a non-vanishing limit of the martingale used therein to the percolation probability. A number of the results derived here are applied in the accompanying paper [A Č19].

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“…More recent developments can be found for instance in [RS13], [PR15], [Szn15], [DPR18b] and [DPR18a]. For the particular case of the Gaussian free field on regular trees we also refer to [Szn16], [Szn19] and [A Č19]; for more general transient trees to [AS18].…”

Section: Introductionmentioning

confidence: 99%

Level-set percolation of the Gaussian free field on regular graphs II: Finite expanders

Abächerli¹,

Černý²

2019

Preprint

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We consider the zero-average Gaussian free field on a certain class of finite dregular graphs for fixed d ≥ 3. 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 exhibits a phase transition at level h , which agrees with the critical value for level-set percolation of the Gaussian free field on the infinite d-regular tree. More precisely, we show that, with probability tending to one as the size of the finite graphs tends to infinity, the level set above level h does not contain any connected component of larger than logarithmic size whenever h > h , and on the contrary, whenever h < h , a linear fraction of the vertices is contained in connected components of the level set above level h having a size of at least a small fractional power of the total size of the graph. It remains open whether in the supercritical phase h < h , as the size of the graphs tends to infinity, one observes the emergence of a (potentially unique) giant connected component of the level set above level h. The proofs in this article make use of results from the accompanying paper [A Č19].

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On coupling and “vacant set level set” percolation

Cited by 6 publications

References 16 publications

Percolation for two-dimensional excursion clouds and the discrete Gaussian free field

Percolation for two-dimensional excursion clouds and the discrete Gaussian free field

Level-set percolation of the Gaussian free field on regular graphs I: Regular trees

Level-set percolation of the Gaussian free field on regular graphs II: Finite expanders

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