Marcelo Hilario, Xinyi Li, Petr Panov
In this work, we prove a shape theorem for Poisson cylinders and give a power law bound on surface fluctuations. We prove that for any $a \in (1/2, 1)$, conditioned on the origin being in the set of cylinders, every point in this set, whose Euclidean norm is less than $R$, lies at an internal distance less than $R+O(R^a)$ from the origin.
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We consider the Poisson cylinder model in $d$-dimensional hyperbolic space. We show that in contrast to the Euclidean case, there is a phase transition in the connectivity of the collection of cylinders as the intensity parameter varies. We also show that for any non-trivial intensity, the diameter of the collection of cylinders is infinite.
We consider the Poisson cylinder model in ${\mathbb R}^d$, $d\ge 3$. We show that given any two cylinders ${\mathfrak c}_1$ and ${\mathfrak c}_2$ in the process, there is a sequence of at most $d-2$ other cylinders creating a connection between ${\mathfrak c}_1$ and ${\mathfrak c}_2$. In particular, this shows that the union of the cylinders is a connected set, answering a question appearing in a previous paper. We also show that there are cylinders in the process that are not connected by a sequence of at most $d-3$ other cylinders. Thus, the diameter of the cluster of cylinders equals $d-2$.
Main characteristics of stationary anisotropic Poisson processes of cylinders (dilated k-dimensional flats) in d-dimensional Euclidean space are studied. Explicit formulae for the capacity functional, the covariance function, the contact distribution function, the volume fraction, and the intensity of the surface area measure are given which can be used directly in applications.
Random union sets $Z$ associated with stationary Poisson processes of $k$-cylinders in $\mathbb{R}^d$ are considered. Under general conditions on the typical cylinder base a concentration inequality for the volume of $Z$ restricted to a compact window is derived. Assuming convexity of the typical cylinder base and isotropy of $Z$ a concentration inequality for intrinsic volumes of arbitrary order is established. A number of special cases are discussed, for example the case when the cylinder bases arise from a random rotation of a fixed convex body. Also the situation of expanding windows is st…
In this paper we deal with the classical problem of random cover times. We investigate the distribution of the time it takes for a Poisson process of cylinders to cover a set $A \subset \mathbb{R}^d.$ This Poisson process of cylinders is invariant under rotations, reflections and translations, and in addition we add a time component so that cylinders are "raining from the sky" at unit rate. Our main results concerns the asymptotic of this cover time as the set $A$ grows. If the set $A$ is discrete and well separated, we show convergence of the cover time to a Gumbel distribution. If instead $A…
Percolation of fat Poisson cylinders in hyperbolic space
本文研究双曲空间中“胖”圆柱的 Poisson 过程，通过建立其与半尺度不变随机分形模型的新联系，证明该模型经历渗流相变，并得出 $\mathbb{R}^d$ 中分形球模型具有非空片相的次要结果。
This paper deals with the union set of a stationary Poisson process of cylinders in $\mathbb{R}^n$ having an $(n-m)$-dimensional base and an $m$-dimensional direction space, where $m\in\{0,1,\ldots,n-1\}$ and $n\geq 2$. The concept simultaneously generalises those of a Boolean model and a Poisson hyperplane or $m$-flat process. Under very general conditions on the typical cylinder base a Berry-Esseen bound for the volume of the union set within a sequence of growing test sets is derived. Assuming convexity of the cylinder bases and of the window a similar result is shown for a broad class of g…
We consider a Poisson point process on the space of lines in R^d, where a multiplicative factor u>0 of the intensity measure determines the density of lines. Each line in the process is taken as the axis of a bi-infinite cylinder of radius 1. We investigate percolative properties of the vacant set, defined as the subset of R^d that is not covered by any such cylinder. We show that in dimensions d >= 4, there is a critical value u_*(d) \in (0,\infty), such that with probability 1, the vacant set has an unbounded component if u<u_*(d), and only bounded components if u>u_*(d). For d=3, we prove t…
Visibility in Brownain interlacements, Poisson cylinders and Boolean models
本文研究了Brownian interlacements等三种模型空置集内的可见性，获得了距离r可见性概率的精确渐近界，发现在$d\geq4$维度下三者标度一致，并改进了已有文献的主要结果。
On the visibility window for Brownian interlacements, Poisson cylinders and Boolean models
本文研究了 $\mathbb R^d$ 中三种空间相关性缓慢衰减模型的空缺集内部可见性，证明了在给定可见条件下，最大可见球半径与可见性窗口之比弱收敛于显式强度的指数分布。
We prove a central limit theorem for the winding number of a directed polymer on a cylinder, which is equivalent with proving the Gaussian fluctuations of the endpoint of the directed polymer in a spatial periodic environment.
We use coupling ideas introduced in \cite{levine2018long} to show that an IDLA process on a cylinder graph $G\times \mathbb{Z}$ forgets a typical initial profile in $\mathcal{O}( N\sqrt{τ_N} (\log \! N)^2 )$ steps for large $N$, where $N$ is the size of the base graph $G$, and $τ_N$ is the total variation mixing time of a simple random walk on $G$. The main new ingredient is a maximal fluctuations bound for IDLA on $G\times \mathbb{Z}$ which only relies on the mixing properties of the base graph $G$ and the Abelian property.
Gaussian fluctuations for Internal DLA on cylinders
研究圆柱体上内部DLA的波动特性，证明其平均波动收敛于GFF，并改进了集群最大波动的界，推导出该几何下IDLA的形状定理。
We study the complementary set of a Poissonian ensemble of infinite cylinders in R^3, for which an intensity parameter u > 0 controls the amount of cylinders to be removed from the ambient space. We establish a non-trivial phase transition, for the existence of an unbounded connected component of this set, as u crosses a critical non-degenerate intensity u*. We moreover show that this complementary set percolates in a sufficiently thick slab, in spite of the fact that it does not percolate in any given plane of R^3, regardless of the choice of u.
We show that space-time averages of the fluctuations of internal DLA on a cylinder Z_N x Z converge in law to the Gaussian free field.
We prove that homogenous sums inside a fixed discrete Poisson chaos are universal with respect to normal approximations. This result parallels some recent findings, in a Gaussian context, by Nourdin, Peccati and Reinert (2010). As a by-product of our analysis, we provide some refinements of the CLTs for random variables on the Poisson space proved by Peccati, Solé, Taqqu and Utzet (2010), and by Peccati and Zheng (2010).
Functional limit theorems are presented for the rescaled occupation time fluctuations process of a critical finite variance branching particle system in $R^d$ with symmetric a-stable motion starting off from either a standard Poisson random field or from the equilibrium distribution for intermediate dimensions a<d<2a. The limit processes are determined sub-fractional and fractional Brownian motion respectively.
We use the periodic Schur process, introduced in arXiv:math/0601019v1, to study the random height function of lozenge tilings (equivalently, dimers) on an infinite cylinder distributed under two variants of the $q^{\operatorname{vol}}$ measure. Under the first variant, corresponding to random cylindric partitions, the height function converges to a deterministic limit shape and fluctuations around it are given by the Gaussian free field in the conformal structure predicted by the Kenyon-Okounkov conjecture. Under the second variant, corresponding to an unrestricted dimer model on the cylinder,…
We study multi-dimensional normal approximations on the Poisson space by means of Malliavin calculus, Stein's method and probabilistic interpolations. Our results yield new multi-dimensional central limit theorems for multiple integrals with respect to Poisson measures -- thus significantly extending previous works by Peccati, Solé, Taqqu and Utzet. Several explicit examples (including in particular vectors of linear and non-linear functionals of Ornstein-Uhlenbeck Lévy processes) are discussed in detail.
We investigate the high-dimensional asymptotic distributional behavior of the $\mathbf{f}$-vector of a random Vietoris-Rips complex, that is generated over a stationary Poisson point process in $[-\frac{1}{2}, \frac{1}{2}]^d$ as the space dimension and the intensity tend to infinity while the radius parameter tends to zero simultaneously.
Continuing the analysis initiated in Lachiéze-Rey and Peccati (2011), we use contraction operators to study the normal approximation of random variables having the form of a U-statistic written on the points in the support of a random Poisson measure. Applications are provided: to boolean models, and coverage of random networks.
Fluctuations of $λ$-geodesic Poisson hyperplanes in hyperbolic space
本文研究d维双曲空间中λ-测地超平面Poisson过程总表面积的涨落，证明λ<1时仅在d=2,3满足定量中心极限定理，d≥4时给出非高斯无穷可分极限分布，而λ=1的极限球面情形对所有d≥2具有极限方差1/2的非标准中心极限定理。
We construct a probabilistic model for the number of divisors of a random uniform integer that converges in the mod-Poisson sense to the same limiting function as its original counterpart, the one arising in the Sathe-Selberg theorem. This construction involves a conditioning and gives an alternative perspective to the usual paradigm of "hybrid product" models developed by Gonek, Hughes and Keating in the case of the Riemann Zeta function.
Limit theorems are presented for the rescaled occupation time fluctuation process of a critical finite variance branching particle system in $\mathbb{R}^{d}$ with symmetric $α$-stable motion starting off from either a standard Poisson random field or the equilibrium distribution for critical $d=2α$ and large $d>2α$ dimensions. The limit processes are generalised Wiener processes. The obtained convergence is in space-time, finite-dimensional distributions sense. With the addtional assumption on the branching law we obtain functional convergence.
We consider a semi-scale invariant version of the Poisson cylinder model which in a natural way induces a random fractal set. We show that this random fractal exhibits an existence phase transition for any dimension $d\geq 2,$ and a connectivity phase transition whenever $d\geq 4.$ We determine the exact value of the critical point of the existence phase transition, and we show that the fractal set is almost surely empty at this critical point. A key ingredient when analysing the connectivity phase transition is to consider a restriction of the full process onto a subspace. We show that this r…
We study the normal approximation of functionals of Poisson measures having the form of a finite sum of multiple integrals. When the integrands are nonnegative, our results yield necessary and sufficient conditions for central limit theorems. These conditions can always be expressed in terms of contraction operators or, equivalently, fourth cumulants. Our findings are specifically tailored to deal with the normal approximation of the geometric $U$-statistics introduced by Reitzner and Schulte (2011). In particular, we shall provide a new analytic characterization of geometric random graphs who…
We prove a Poisson limit theorem in the total variation distance of functionals of a general Poisson point process using the Malliavin-Stein method. Our estimates only involve first and second order difference operators and are closely related to the corresponding bounds for the normal approximation in the Wasserstein distance by Last, Peccati and Schulte (2016). As an application of this Poisson limit theorem, we consider a stationary Poisson point process in $\mathbb{R}^d$ and connect any two points whenever their distance is less than or equal to a prescribed distance parameter. This constr…
Fluctuations of the giant of Poisson random graphs
本文研究泊松随机图巨分量的涨落，将动态 Erdős-Rényi 随机图的过程级高斯涨落结果推广到经验权重分布及其二阶矩收敛的秩一非齐次模型。
The classical Poisson theorem says that if $ξ_1,ξ_2,...$ are i.i.d. 0--1 Bernoulli random variables taking on 1 with probability $p_n\equiv \la/n$ then the sum $S_n=\sum_{i=1}^nξ_i$ is asymptotically in $n$ Poisson distributed with the parameter $\la$. It turns out that this result can be extended to sums of the form $S_n=\sum_{i=1}^nξ_{q_1(i)}... ξ_{q_\ell(i)}$ where now $p_n\equiv(\la/n)^{1/\ell}$ and $1\leq q_1(i) <... <q_\ell(i)$ are integer valued increasing functions. We obtain also Poissonian limit for numbers of arrivals to small sets of $\ell$-tuples $X_{q_1(i)},...,X_{q_\ell(i)}$ for…
Domino tilings of black-and-white Temperleyan cylinders
研究了黑白Temperley圆柱的多米诺铺砌，证明了高度波动收敛于高斯自由场并出现离散高斯分布。
We prove that first-passage percolation times across thin cylinders of the form $[0,n]\times [-h_n,h_n]^{d-1}$ obey Gaussian central limit theorems as long as $h_n$ grows slower than $n^{1/(d+1)}$. It is an open question as to what is the fastest that $h_n$ can grow so that a Gaussian CLT still holds. Under the natural but unproven assumption about existence of fluctuation and transversal exponents, and strict convexity of the limiting shape in the direction of $(1,0,...,0)$, we prove that in dimensions 2 and 3 the CLT holds all the way up to the height of the unrestricted geodesic. We also pr…
On the maximum visibility in a ball through the vacant set of Poissonian obstacles
本文研究三种障碍物模型中球内最大可见距离，证明其除以特定对数因子后依概率收敛于显式常数，并给出维度3时布朗交错的修正因子。
Multi-hop visibility through the vacant set of Poissonian obstacles
该研究针对d维欧几里得空间中三种不同障碍物模型的空集，推导了多跳可见性概率的精确界，观察到其在两个特征尺度上的普适行为，推广了早期针对单跳的相关结果。
A fluctuation theory and, in particular, a theory of scale functions is developed for upwards skip-free Lévy chains, i.e. for right-continuous random walks embedded into continuous time as compound Poisson processes. This is done by analogy to the spectrally negative class of Lévy processes -- several results, however, can be made more explicit/exhaustive in our compound Poisson setting. In particular, the scale functions admit a linear recursion, of constant order when the support of the jump measure is bounded, by means of which they can be calculated -- some examples are considered.
Consider a stationary Poisson point process in $\mathbb{R}^d$ and connect any two points whenever their distance is less than or equal to a prescribed distance parameter. This construction gives rise to the well known random geometric graph. The number of edges of this graph is counted that have midpoint in the $d$-dimensional unit ball. A quantitative central limit theorem for this counting statistic is derived, as the space dimension $d$ and the intensity of the Poisson point process tend to infinity simultaneously.
Occupation time fluctuation limits of particle systems in R^d with independent motions (symmetric stable Levy process, with or without critical branching) have been studied assuming initial distributions given by Poisson random measures (homogeneous and some inhomogeneous cases). In this paper, with d=1 for simplicity, we extend previous results to a wide class of initial measures obeying a quasi-homogeneity property, which includes as special cases homogeneous Poisson measures and many deterministic measures (simple example: one atom at each point of Z), by means of a new unified approach. In…
Occupation-Time Fluctuations of an Age-Dependent Branching System Driven by Poisson Immigration
研究临界年龄依赖分支粒子系统在泊松移民驱动下的占据时间波动，证明在适当时空尺度下，重标度波动弱收敛到具有明确协方差结构的高斯过程，该过程自相似且具有长程依赖性。
Thresholds and Fluctuations for Colorful Arithmetic Progressions in Sparse Random Colorings
本文确定了稀疏随机着色中彩色等差数列出现的完整阈值区域，并刻画了其数量在可满足区域内的渐近正态性及阈值曲面上的三种泊松型波动状态。
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