Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0410336

The critical probability for random Voronoi percolation in the plane is 1/2

Bela Bollobas·Oliver Riordan
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study percolation in the following random environment: let $Z$ be a Poisson process of constant intensity in the plane, and form the Voronoi tessellation of the plane with respect to $Z$. Colour each Voronoi cell black with probability $p$, independently of the other cells. We show that the critical probability is 1/2. More precisely, if $p>1/2$ then the union of the black cells contains an infinite component with probability 1, while if $p<1/2$ then the distribution of the size of the component of black cells containing a given point decays exponentially. These results are analogous to Kesten's results for bond percolation in the square lattice. The result corresponding to Harris' Theorem for bond percolation in the square lattice is known: Zvavitch noted that one of the many proofs of this result can easily be adapted to the random Voronoi setting. For Kesten's results, none of the existing proofs seems to adapt. The methods used here also give a new and very simple proof of Kesten's Theorem for the square lattice; we hope they will be applicable in other contexts as well.

Explore related subjects

math.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Bela Bollobas, Oliver Riordan. 2005-09-27. The critical probability for random Voronoi percolation in the plane is 1/2. https://doi.org/10.1007/s00440-005-0490-z

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Breuer-Major Theorems for Hilbert Space-Valued Random Variables

Let $\{X_k\}_{k\in\mathbb Z}$ be a stationary Gaussian process with values in a separable Hilbert space $\mathcal H_1$, and let $G:\mathcal H_1\to\mathcal H_2$ be a measurable map into another separable Hilbert space $\mathcal H_2$. We derive a central limit theorem for the centered normalized partial sums of the Hilbert space-valued subordinated process $\{G[X_k]\}_{k\in\mathbb Z}$. Our result holds under either of two sets of sufficient conditions, formulated in terms of the transformation $G$ and the temporal and cross-sectional dependence structure of $\{X_k\}_{k\in\mathbb Z}$. These conditions coincide in finite dimensions but lead to genuinely different phenomena in the infinite-dimensional setting. The proof relies on the recently developed Fourth Moment Theorem on Hilbert spaces, leveraging tools from the infinite-dimensional Malliavin-Stein framework. We also provide continuous-time and quantitative versions of the central limit theorem. In a series of examples, we recover and strengthen limit theorems for a wide array of statistics relevant in functional data analysis, and present, as an application of our result, a novel limit theorem in the framework of neural operators.

math.PR↗
arXiv2026

The Fourth-Moment Theorem on Hilbert Spaces

In this work, we establish conditions ensuring convergence in distribution of a sequence admitting a Wiener-Itô chaos representation to a Gaussian measure on a separable Hilbert space. Our first main result shows that, assuming convergence of the associated covariance operators in the trace-class norm, a sequence lying in a fixed Wiener-Itô chaos converges in distribution if and only if its fourth weak moments converge to the corresponding Gaussian moments. For general sequences with infinite chaos expansions, we derive analogous sufficient conditions for convergence in distribution. A key ingredient in our approach is a Stein-Malliavin bound formulated with respect to a distance that metrizes weak convergence of probability measures on separable Hilbert spaces. The results are infinite-dimensional extensions of the classical real-valued Fourth-Moment Theorem of Nualart and Peccati [Ann. Probab. 33, 177-193 (2005)]. Our work builds upon the work by Bourguin and Campese [Electron. J. Probab. 25, 1-30 (2020)] who claimed a Fourth-Moment Theorem in separable Hilbert spaces. However, a recent work by Bassetti, Bourguin, Campese, and Peccati [Stat. Probab. Lett. 233, 110671, (2026)] showed that the distance employed in the former article does not metrize weak convergence of probability measures on separable Hilbert spaces. Consequently, the conditions stated in Bourguin and Campese are not sufficient to recover a valid Fourth-Moment Theorem in the Hilbert-space setting.

math.PR↗
arXiv2026

Collision types and times in interacting particle systems

We consider a system of stochastic interacting particles with general diffusion coefficient and drift functions and we study the types of collisions that arise in them. In particular, interactions between particles are inversely proportional to their separation, and the coupling function of interaction is also considered in great generality. Our main result shows that, under positivity and the stated variance compatibility conditions, no two distinct positive-root hyperplanes are reached simultaneously at a positive time. In type $A_{N-1}$, this excludes both collisions involving three or more particles and simultaneous collisions of disjoint pairs. In order to obtain our results we make use of symmetric polynomials in squared root projections; the degree of these polynomials indicates the type of collision, and by a locality argument we show that polynomials indicating a non-simple collision almost surely do not cancel. We use this result to obtain upper and lower bounds for the Hausdorff dimension of the set of collision times in terms of interaction-to-variance ratios. These bounds coincide in particular constant-ratio cases. Our results cover many of the most well-known particle systems, such as the Dyson model and Wishart processes and their extensions to non-constant diffusion coefficients and background drifts.

math.PR↗