The Sign Clusters of the Massless Gaussian Free Field Percolate on 
                
                  
                
                $${\mathbb{Z}^{d}, d \geqslant 3}$$
                
                  
                    
                      
                        
                          Z
                        
                        d
                      
                      ,
                      d
                      ⩾
                      3

Drewitz, Alexander; Prévost, Alexis; Rodriguez, Pierre-Françcois

doi:10.1007/s00220-018-3209-6

Cited by 24 publications

(1 citation statement)

References 23 publications

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“…It has attracted attention for a long time, dating back to [7, 21, 24]. 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

\mathbb {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

\mathbb {Z}^d

in [13].…”

Section: Introductionmentioning

confidence: 99%

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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\mathbb {Z}^d

\mathbb {Z}^d

in [13].…”

Section: Introductionmentioning

confidence: 99%

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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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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Analyticity of Gaussian Free Field Percolation Observables

Panagiotis

Severo

2022

Commun. Math. Phys.

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We prove that cluster observables of level-sets of the Gaussian free field on the hypercubic lattice $${{\mathbb {Z}}}^d$$ Z d , $$d\ge 3$$ d ≥ 3 , are analytic on the whole off-critical regime $${{\mathbb {R}}}\setminus \{h_*\}$$ R \ { h ∗ } . This result concerns in particular the percolation density function $$\theta (h)$$ θ ( h ) and the (truncated) susceptibility $$\chi (h)$$ χ ( h ) . As an important step towards the proof, we show the exponential decay in probability for the capacity of a finite cluster for all $$h\ne h_*$$ h ≠ h ∗ , which we believe to be a result of independent interest. We also discuss the case of general transient graphs.

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The Sign Clusters of the Massless Gaussian Free Field Percolate on $${\mathbb{Z}^{d}, d \geqslant 3}$$ Z d , d ⩾ 3

Cited by 24 publications

References 23 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

Disconnection and Entropic Repulsion for the Harmonic Crystal with Random Conductances

Analyticity of Gaussian Free Field Percolation Observables

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