It is shown that a large class of events in a product probability space are highly sensitive to noise, in the sense that with high probability, the configuration with an arbitrary small percent of random errors gives almost no prediction whether the event occurs. On the other hand, weighted majority functions are shown to be noise-stable. Several necessary and sufficient conditions for noise sensitivity and stability are given.
Consider, for example, bond percolation on an $n+1$ by $n$ grid. A configuration is a function that assigns to every edge the value 0 or 1. Let $ω$ be a random configuration, selected according to the uniform measure. A crossing is a path that joins the left and right sides of the rectangle, and consists entirely of edges $e$ with $ω(e)=1$. By duality, the probability for having a crossing is 1/2. Fix an $ε\in(0,1)$. For each edge $e$, let $ω'(e)=ω(e)$ with probability $1-ε$, and $ω'(e)=1-ω(e)$ with probability $ε$, independently of the other edges. Let $p(τ)$ be the probability for having a crossing in $ω$, conditioned on $ω'=τ$. Then for all $n$ sufficiently large, $P\{τ: |p(τ)-1/2|>ε\}<ε$.
We prove that the Poisson-Boolean percolation on $\mathbb{R}^d$ undergoes a sharp phase transition in any dimension under the assumption that the radius distribution has a $5d-3$ finite moment (in particular we do not assume that the distribution is bounded). More precisely, we prove that: -In the whole subcritical regime, the expected size of the cluster of the origin is finite, and furthermore we obtain bounds for the origin to be connected to distance $n$: when the radius distribution has a finite exponential moment, the probability decays exponentially fast in $n$, and when the radius dist…
We investigate percolation in the Boolean model with convex grains in high dimension. For each dimension d, one fixes a compact, convex and symmetric set K $\subset$ R d with non empty interior. In a first setting, the Boolean model is a reunion of translates of K. In a second setting, the Boolean model is a reunion of translates of K or $ρ$K for a further parameter $ρ$ $\in$ (1, 2). We give the asymptotic behavior of the percolation probability and of the percolation threshold in the two settings.
We consider a multiscale Boolean percolation on $\mathbb R^d$ with radius distribution $μ$ on $[1,+\infty)$, $d\ge 2$. The model is defined by superposing the original Boolean percolation model with radius distribution $μ$ with a countable number of scaled independent copies. The $n$-th copy is a Boolean percolation with radius distribution $μ|_{[1,κ]}$ rescaled by $κ^{n}$. We prove that under some regularity assumption on $μ$, the subcritical phase of the multiscale model is sharp for $κ$ large enough. Moreover, we prove that the existence of an unbounded connected component depends only on t…
We consider the discrete Boolean model of percolation on graphs satisfying a doubling metric condition. We study sufficient conditions on the distribution of the radii of balls placed at the points of a Bernoulli point process for the absence of percolation, provided that the retention parameter of the underlying point process is small enough. We exhibit three families of interesting graphs where the main result of this work holds. Finally, we give sufficient conditions for ergodicity of the discrete Boolean model of percolation.
It is well known that there are two regimes in a standard one-dimensional Boolean percolation model: either the entire space is covered a.s., or the covered volume fraction is strictly less than one. The aim of this work is to demonstrate that there is a third possibility in a Boolean model with totally asymmetric grains on a half-line: a.s. there is no unbounded component, but the covered volume fraction is one. An explicit condition is given characterizing the existence of an unbounded occupied component.
We consider the Boolean model with random radii based on Cox point processes. Under a condition of stabilization for the random environment, we establish existence and non-existence of subcritical regimes for the size of the cluster at the origin in terms of volume, diameter and number of points. Further, we prove uniqueness of the infinite cluster for sufficiently connected environments.
We consider Poisson Boolean percolation on $\mathbb R^d$ with power-law distribution on the radius with a finite $d$-moment for $d\ge 2$. We prove that subcritical sharpness occurs for all but a countable number of power-law distributions. This extends the results of Duminil-Copin--Raoufi--Tassion where subcritical sharpness is proved under the assumption that the radii distribution has a $5d-3$ finite moment. Our proofs techniques are different from their paper: we do not use randomized algorithm and rely on specific independence properties of Boolean percolation, inherited from the underlyin…
We establish the mean-field bounds $γ\ge 1$, $δ\ge 2$ and $\triangle \ge 2$ on the critical exponents of the Poisson-Boolean continuum percolation model under a moment condition on the radii; these were previously known only in the special case of fixed radii (in the case of $γ$), or not at all (in the case of $δ$ and $\triangle$). We deduce these as consequences of the mean-field bound $β\le 1$, recently established by Duminil-Copin, Raoufi and Tassion under the same moment condition, using a relative entropy method introduced by the authors in previous work.
Disagreement percolation connects a Gibbs lattice gas and i.i.d. site percolation on the same lattice such that non-percolation implies uniqueness of the Gibbs measure. This work generalises disagreement percolation to the hard-sphere model and the Boolean model. Non-percolation of the Boolean model implies the uniqueness of the Gibbs measure and exponential decay of pair correlations and finite volume errors. Hence, lower bounds on the critical intensity for percolation of the Boolean model imply lower bounds on the critical activity for a (potential) phase transition. These lower bounds impr…
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, Saks, Schramm and Servedio (2005)) 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 …
We consider the Boolean model on $\R^d$. We prove some equivalences between subcritical percolation properties. Let us introduce some notations to state one of these equivalences. Let $C$ denote the connected component of the origin in the Boolean model. Let $|C|$ denotes its volume. Let $\ell$ denote the maximal length of a chain of random balls from the origin. Under optimal integrability conditions on the radii, we prove that $E(|C|)$ is finite if and only if there exists $A,B >0$ such that $¶(\ell \ge n) \le Ae^{-Bn}$ for all $n \ge 1$.
Consider critical Bernoulli percolation in the plane. We give a new proof of the sharp noise sensitivity theorem shown by Garban, Pete and Schramm. Contrary to the previous approaches, we do not use any spectral tool. We rather study differential inequalities satisfied by a dynamical four-arm event, in the spirit of Kesten's proof of scaling relations. We also obtain new results in dynamical percolation. In particular, we prove that the Hausdorff dimension of the set of times with both primal and dual percolation equals $2/3$ a.s.
We state and prove a version for invasion percolation of Schramm and Smirnov's (2001) discrete gluing theorem for critical planar percolation. The result is a first step toward establishing factorization and characterizing noise sensitivity for invasion percolation.
Detection, coverage and percolation in dynamic Boolean models with random radii based on $α$-stable processes
School of Mathematics, University of Leeds（利兹大学数学学院）；Technische Universität Braunschweig（布伦瑞克工业大学）；Weierstrass Institute for Applied Analysis and Stochastics（魏尔斯特拉斯应用分析与随机过程研究所）
In this paper we study noise sensitivity and threshold phenomena for Poisson Voronoi percolation on $\mathbb{R}^2$. In the setting of Boolean functions, both threshold phenomena and noise sensitivity can be understood via the study of randomized algorithms. Together with a simple discretization argument, such techniques apply also to the continuum setting. Via the study of a suitable algorithm we show that box-crossing events in Voronoi percolation are noise sensitive and present a threshold phenomenon with polynomial window. We also study the effect of other kinds of perturbations, and emphas…
For a given dimension d $\ge$ 2 and a finite measure $ν$ on (0, +$\infty$), we consider $ξ$ a Poisson point process on R d x (0, +$\infty$) with intensity measure dc $\otimes$ $ν$ where dc denotes the Lebesgue measure on R d. We consider the Boolean model $Σ$ = $\cup$ (c,r)$\in$$ξ$ B(c, r) where B(c, r) denotes the open ball centered at c with radius r. For every x, y $\in$ R d we define T (x, y) as the minimum time needed to travel from x to y by a traveler that walks at speed 1 outside $Σ$ and at infinite speed inside $Σ$. By a standard application of Kingman sub-additive theorem, one easily…
We show that planar Bargmann-Fock percolation is noise sensitive under the Ornstein-Ulhenbeck process. The proof is based on the randomized algorithm approach introduced by Schramm and Steif and gives quantitative polynomial bounds on the noise sensitivity of crossing events for Bargmann-Fock. A rather counter-intuitive consequence is as follows. Let $F$ be a Bargmann-Fock Gaussian field in $\mathbb{R}^3$ and consider two horizontal planes $P_1,P_2$ at small distance $\varepsilon$ from each other. Even though $F$ is a.s. analytic, the above noise sensitivity statement implies that the full res…
Mobile Geometric Graphs: Detection, Coverage and Percolation
我们考虑由van den Berg，Meester和White（1997）引入的以下动态布尔模型。在时刻0，设图的节点是R^d中强度恒定的泊松点过程，并让每个节点按照布朗运动独立运动。在任意时刻t，我们在每对节点之间放置一条边，如果它们的距离至多为r。我们在这个模型中研究了三个特征：检测（直到目标点-固定或移动-距离图的某个节点在距离r以内的时间），覆盖（直到有限盒内的所有点都被图检测到的时间）和渗透（直到给定节点属于图的无限连通分量的时间）。我们获得精确的渐近这些功能相结合的想法，从随机几何，耦合和多尺度分析。
We study functions on the infinite-dimensional Hamming cube $\{-1,1\}^\infty$, in particular Boolean functions into $\{-1,1\}$, generalising results on analysis of Boolean functions on $\{-1,1\}^n$ for $n\in\mathbb{N}$. The notion of noise sensitivity, first studied in arXiv:math/9811157 , is extended to this setting, and basic Fourier formulas are established. We also prove hypercontractivity estimates for these functions, and give a version of the Kahn-Kalai-Linial theorem giving a bound relating the total influence to the maximal influence. Particular attention is paid to so-called finitary…
The estimates for Boolean surface area of polynomial threshold functions
Nankai University（南开大学）；Institute for Basic Science (IBS)（基础科学研究院）；University of Washington（华盛顿大学）；MSU（密歇根州立大学）；Max Planck Institute for Mathematics（马普数学研究所）；University of South Carolina（南卡罗来纳大学）
Bootstrap percolation is a wide class of monotone cellular automata with random initial state. In this work we develop tools for studying in full generality one of the three `universality' classes of bootstrap percolation models in two dimensions, termed subcritical. We introduce the new notion of `critical densities' serving the role of `difficulties' for critical models, but adapted to subcritical ones. We characterise the critical probability in terms of these quantities and successfully apply this link to prove new and old results for concrete models such as DTBP and Spiral as well as a ge…
In a recent paper by Jonasson and Steif, definitions to describe the volatility of sequences of Boolean functions, \( f_n \colon \{ -1,1 \}^n \to \{ -1,1 \} \) were introduced. We continue their study of how these definitions relate to noise stability and noise sensitivity. Our main results are that the set of volatile sequences of Boolean functions is a natural way "dense" in the set of all sequences of Boolean functions, and that the set of non-volatile Boolean sequences is not "dense" in the set of noise stable sequences of Boolean functions.
