Equality of critical parameters for percolation of Gaussian free field level sets

Duminil-Copin, Hugo; Goswami, Subhajit; Rodriguez, Pierre-François; Severo, Franco

doi:10.1215/00127094-2022-0017

Cited by 18 publications

(16 citation statements)

References 68 publications

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“…The diverging dimension is important for the arguments therein, since the correlations of the field decay with the dimension (as 𝑑 −1 for the neighbouring vertices). Several sprinkling steps are also used in the recent paper [15], which proves the sharpness of the phase transition for the level set of the Gaussian free field on ℤ 𝑑 , 𝑑 ≥ 3. Note also that the results of [15] can be combined with [1] to show the existence of the giant component for the supercritical level set of the zero-average Gaussian free field on a large discrete torus (cf.…”

Section: Introductionmentioning

confidence: 92%

“…It is the main ingredient of our sprinkling construction, as described in the introduction, but also will be used in Section 4 to construct a new coupling of Ψ  𝑛 and 𝜑 𝕋 𝑑 . The construction of this decomposition is inspired by a similar decomposition for the usual Gaussian free field on ℤ 𝑑 from [15], see Lemma 3.1 therein. However, the zero-average property introduces certain complications making the decomposition less straightforward.…”

Section: Decomposition Of the Fieldmentioning

confidence: 99%

“…In the last decade, it has been subject to intensive research after a non-trivial percolation phase transition has been identified for this model on ℤ 𝑑 in [27], see for instance [2,9,12,29]. Only very recently, in the remarkable paper [15], it has been shown that this phase transition is sharp, and, rather amazingly, the critical exponents have been identified for a related model of the level-set percolation on the cable system of ℤ 𝑑 in [13].…”

Section: Introductionmentioning

confidence: 99%

“…Note also that the results of [15] can be combined with [1] to show the existence of the giant component for the supercritical level set of the zero-average Gaussian free field on a large discrete torus (cf. [15,Section 1.2]).…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we thus develop a new decomposition of the zero-average Gaussian free field which provides enough independence to be useful in a sprinkling argument and which is of independent interest, see Section 3. It is inspired by a similar decomposition of the (usual) Gaussian free field on ℤ 𝑑 from [15]. Using this decomposition we will write Ψ  𝑛 as a sum of two independent components…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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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Section: Introductionmentioning

confidence: 92%

Section: Decomposition Of the Fieldmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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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On the cost of the bubble set for random interlacements

Sznitman

2023

Invent. math.

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The main focus of this article concerns the strongly percolative regime of the vacant set of random interlacements on ${\mathbb{Z}}^{d}$ Z d , $d \ge 3$ d ≥ 3 . We investigate the occurrence in a large box of an excessive fraction of sites that get disconnected by the interlacements from the boundary of a concentric box of double size. The results significantly improve our findings in Sznitman (Probab. Math. Phys. 2–3:563–311, 2021). In particular, if, as expected, the critical levels for percolation and for strong percolation of the vacant set of random interlacements coincide, the asymptotic upper bound that we derive here matches in principal order a previously known lower bound. A challenging difficulty revolves around the possible occurrence of droplets that could get secluded by the random interlacements and thus contribute to the excess of disconnected sites, somewhat in the spirit of the Wulff droplets for Bernoulli percolation or for the Ising model. This feature is reflected in the present context by the so-called bubble set, a possibly quite irregular random set. A pivotal progress in this work has to do with the improved construction of a coarse grained random set accounting for the cost of the bubble set. This construction heavily draws both on the method of enlargement of obstacles originally developed in the mid-nineties in the context of Brownian motion in a Poissonian potential in Sznitman (Ann. Probab. 25(3):1180–1209, 1997; Brownian Motion, Obstacles and Random Media. Springer, Berlin, 1998), and on the resonance sets recently introduced by Nitzschner and Sznitman in (J. Eur. Math. Soc. 22(8):2629–2672, 2020) and further developed in a discrete set-up by Chiarini and Nitzschner in (Commun. Math. Phys. 386(3):1685–1745, 2021).

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Geometry of Gaussian free field sign clusters and random interlacements

Drewitz,

Prévost,

Rodriguez

2024

Probab. Theory Relat. Fields

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For a large class of amenable transient weighted graphs G, we prove that the sign clusters of the Gaussian free field on G fall into a regime of strong supercriticality, in which two infinite sign clusters dominate (one for each sign), and finite sign clusters are necessarily tiny, with overwhelming probability. Examples of graphs belonging to this class include regular lattices such as $${\mathbb {Z}}^d$$ Z d , for $$d\ge 3$$ d ≥ 3 , but also more intricate geometries, such as Cayley graphs of suitably growing (finitely generated) non-Abelian groups, and cases in which random walks exhibit anomalous diffusive behavior, for instance various fractal graphs. As a consequence, we also show that the vacant set of random interlacements on these objects, introduced by Sznitman (Ann Math 171(3):2039–2087, 2010), and which is intimately linked to the free field, contains an infinite connected component at small intensities. In particular, this result settles an open problem from Sznitman (Invent Math 187(3):645–706, 2012).

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Equality of critical parameters for percolation of Gaussian free field level sets

Cited by 18 publications

References 68 publications

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

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

On the cost of the bubble set for random interlacements

Geometry of Gaussian free field sign clusters and random interlacements

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