Percolation of the excursion sets of planar symmetric shot noise fields

Muirhead, Stephen

doi:10.1016/j.spa.2022.01.013

Cited by 5 publications

(4 citation statements)

References 25 publications

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“…We first observe that the finite‐dimensional nature of the original OSSS and Schramm–Steif inequalities has not been—in general—a major obstacle for applying them to random geometric models based on infinite‐dimensional inputs. Indeed, using suitable discretization schemes and some extra technical work, these estimates have been successfully applied to geometric models based on stationary Euclidean Poisson point processes, see [2, 3, 24–26, 30, 45]. This fact notwithstanding, the applications developed in the present paper indicate that the our intrinsic approach is a new valuable tool for establishing quantitative noise sensitivity, existence of exceptional times and sharp phase transition in (possibly dynamical) continuum percolation models, under minimal assumptions and by using strategies of proofs that only require one to prove nondegenaracy of some class of functionals and decay on arm probabilities.…”

Section: Introductionmentioning

confidence: 86%

“…More recently, in [45] RSW‐type estimates, nondegeneracy of crossing events, decay of arm‐probabilities and sharp phase transition were proven. In particular, [45, Proposition 4.2] is proved using the discrete OSSS inequality and our continuum analogue of OSSS inequality can again be used to avoid discretization. Further, one would expect to deduce noise sensitivity and exceptional times for crossings in this model at criticality analogous to Corollary 8.5.…”

Section: Further Applicationsmentioning

confidence: 96%

“…Level set percolation: Percolation of level sets of Poissonian shot-noise random fields has been another model that has received some attention (see [8,16,55]). More recently, in [45] RSW-type estimates, nondegeneracy of crossing events, decay of arm-probabilities and sharp phase transition were proven. In particular, [45, Proposition 4.2] is proved using the discrete OSSS inequality and our continuum analogue of OSSS inequality can again be used to avoid discretization.…”

mentioning

confidence: 97%

“…The above theorem has reduced the proof of noise sensitivity, exceptional times and sharp phase transition for crossing events at criticality in the k-percolation model to showing nondegeneracy of the crossing events and bounds for one-arm probabilities or nonpercolation at criticality. Showing these properties is a seperate percolation theoretic question, often model-specific and in the planar case, these have been achieved in some models via RSW-type estimates (see [4,7,41,45,66,67,74]). See Corollary 8.5 below for a case in which the above estimates are known.…”

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confidence: 99%

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Phase transitions and noise sensitivity on the Poisson space via stopping sets and decision trees

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Peccati

Yogeshwaran

2023

Random Struct Algorithms

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Proofs of sharp phase transition and noise sensitivity in percolation have been significantly simplified by the use of randomized algorithms, via the OSSS inequality (proved by O'Donnell et al. and the Schramm–Steif inequality for the Fourier‐Walsh coefficients of functions defined on the Boolean hypercube. In this article, we prove intrinsic versions of the OSSS and Schramm–Steif inequalities for functionals of a general Poisson process, and apply these new estimates to deduce sufficient conditions—expressed in terms of randomized stopping sets—yielding sharp phase transitions, quantitative noise sensitivity, exceptional times and bounds on critical windows for monotonic Boolean Poisson functions. Our analysis is based on a new general definition of “stopping set”, not requiring any topological property for the underlying measurable space, as well as on the new concept of a “continuous‐time decision tree”, for which we establish several fundamental properties. We apply our findings to the k$$ k $$‐percolation of the Poisson Boolean model and to the Poisson‐based confetti percolation with bounded random grains. In these two models, we reduce the proof of sharp phase transitions for percolation, and of noise sensitivity for crossing events, to the construction of suitable randomized stopping sets and the computation of one‐arm probabilities at criticality. This enables us to settle an open problem suggested by Ahlberg et al. (a special case of which was conjectured earlier by Ahlberg et al. on noise sensitivity of crossing events for the planar Poisson Boolean model with random balls whose radius distribution has finite false(2+αfalse)$$ \left(2+\alpha \right) $$‐moments and also show the same for planar confetti percolation model with bounded random balls. We also prove that critical probability is 1false/2$$ 1/2 $$ for the planar confetti percolation model with bounded, πfalse/2$$ \pi /2 $$‐rotation invariant and reflection invariant random grains. Such a result was conjectured by Benjamini and Schramm in the case of fixed balls and proved by Müller, Hirsch and Ghosh and Roy in the case of balls, boxes and random boxes, respectively; our results contain all previous findings as special cases.

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Section: Introductionmentioning

confidence: 86%

Section: Further Applicationsmentioning

confidence: 96%

mentioning

confidence: 97%

mentioning

confidence: 99%

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Phase transitions and noise sensitivity on the Poisson space via stopping sets and decision trees

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Peccati

Yogeshwaran

2023

Random Struct Algorithms

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Quantitative two-scale stabilization on the Poisson space

Peccati

Yang

2022

Ann. Appl. Probab.

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We establish inequalities for assessing the distance between the distribution of a (possibly multidimensional) functional of a Poisson random measure and that of a Gaussian element. Our bounds only involve add-one cost operators at the order one -that we evaluate and compare at two different scales -and are specifically tailored for studying the Gaussian fluctuations of sequences of geometric functionals displaying a form of weak stabilizationsee Penrose and Yukich (2001) and Penrose (2005). Our main bounds extend the estimates recently exploited by Chatterjee and Sen (2017) in the proof of a quantitative version of the central limit theorem (CLT) for the length of the Poisson-based Euclidean minimal spanning tree (MST). We develop in full detail three applications of our bounds, namely: (i) to a quantitative multidimensional spatial CLT for functionals of the on-line nearest neighbour graph, (ii) to a quantitative multidimensional CLT involving functionals of the empirical measure associated with the edge-length of the Euclidean MST, and (iii) to a collection of multidimensional CLTs for geometric functionals of the excursion set of heavy-tailed shot noise random fields. Application (i) is based on a collection of general probabilistic approximations for strongly stabilizing functionals, that is of independent interest.

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Asymptotics for the critical level and a strong invariance principle for high intensity shot noise fields

Lachièze-Rey,

Muirhead

2023

Ann. Inst. H. Poincaré Probab. Statist.

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Percolation of the excursion sets of planar symmetric shot noise fields

Cited by 5 publications

References 25 publications

Phase transitions and noise sensitivity on the Poisson space via stopping sets and decision trees

Phase transitions and noise sensitivity on the Poisson space via stopping sets and decision trees

Quantitative two-scale stabilization on the Poisson space

Asymptotics for the critical level and a strong invariance principle for high intensity shot noise fields

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