Gady Kozma, Zvi Lotker, Gideon Stupp
For a measure mu supported on a compact connected subset of a Euclidean space which satisfies a uniform d-dimensional decay of the volume of balls we show that the maximal edge in the minimum spanning tree of n indepndent samples from mu is, with high probability (log n/n)^(1/d).
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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…
In this paper we study the existence phase transition of the random fractal ball model and the random fractal box model. We show that both of these are in the empty phase at the critical point of this phase transition.
In this paper we study the existence phase transition of scale invariant random fractal models. We determine the exact value of the critical point of this phase transition for all models satisfying some weak assumptions. In addition, we show that for a large subclass, the fractal model is in the empty phase at the critical point. This subclass of models includes the scale invariant Poisson Boolean model and the Brownian loop soup. In contrast to earlier results in the literature, we do not need to restrict our attention to random fractal models generated by open sets.
Partially motivated by the desire to better understand the connectivity phase transition in fractal percolation, we introduce and study a class of continuum fractal percolation models in dimension d greater than or equal to 2. These include a scale invariant version of the classical (Poisson) Boolean model of stochastic geometry and (for d=2) the Brownian loop soup introduced by Lawler and Werner. The models lead to random fractal sets whose connectivity properties depend on a parameter lambda. In this paper we mainly study the transition between a phase where the random fractal sets are total…
In this paper, we construct a new family of random series defined on $\R^D$, indexed by one scaling parameter and two Hurst-like exponents. The model is close to Takagi-Knopp functions, save for the fact that the underlying partitions of $\R^D$ are not the usual dyadic meshes but random Voronoi tessellations generated by Poisson point processes. This approach leads us to a continuous function whose random graph is shown to be fractal with explicit and equal box and Hausdorff dimensions. The proof of this main result is based on several new distributional properties of the Poisson-Voronoi tesse…
We introduce a new way to sample inhomogeneous random graphs designed to have a lot of flexibility in the assignment of the degree sequence and the individual edge probabilities while remaining tractable. To achieve this we run a Poisson point process over the square $[0,1]^2$, with an intensity proportional to a kernel $W(x,y)$ and identify every couple of vertices of the graph with a subset of the square, adding an edge between them if there is a point in such subset. This ensures unconditional independence among edges and makes many statements much easier to prove in this setting than in ot…
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。
Continuous time branching models are used to create random fractals in a Euclidean space, whose Hausdorff dimension is controlled by an input parameter. Finite realizations are applied in modelling the set of sites visited in models of human and animal mobility.
We study an inhomogeneous random connection model in the connectivity regime. The vertex set of the graph is a homogeneous Poisson point process $\mathcal{P}_s$ of intensity $s>0$ on the unit cube $S=\left(-\frac{1}{2},\frac{1}{2}\right]^{d},$ $d \geq 2$ . Each vertex is endowed with an independent random weight distributed as $W$, where $P(W>w)=w^{-β}1_{[1,\infty)}(w)$, $β>0$. Given the vertex set and the weights an edge exists between $x,y\in \mathcal{P}_s$ with probability $\left(1 - \exp\left( - \frac{ηW_xW_y}{\left(d(x,y)/r\right)^α} \right)\right),$ independent of everything else, where …
The objective of this study is to examine the asymptotic behavior of Betti numbers of Čech complexes treated as stochastic processes and formed from random points in the $d$-dimensional Euclidean space $\mathbb{R}^d$. We consider the case where the points of the Čech complex are generated by a Poisson process with intensity $nf$ for a probability density $f$. We look at the cases where the behavior of the connectivity radius of Čech complex causes simplices of dimension greater than $k+1$ to vanish in probability, the so-called sparse and Poisson regimes, as well when the connectivity radius i…
IDS for subordinate Brownian motions in Poisson random environment on nested fractals
本文研究了在嵌套分形上带有泊松随机环境的从属布朗运动的IDS奇异性，通过将随机势能转化为合金型势能，拓展了伯恩斯坦函数的应用范围。
We consider a class of point processes (pp), which we call {\em sub-Poisson}; these are pp that can be directionally-convexly ($dcx$) dominated by some Poisson pp. The $dcx$ order has already been shown useful in comparing various point process characteristics, including Ripley's and correlation functions as well as shot-noise fields generated by pp, indicating in particular that smaller in the $dcx$ order processes exhibit more regularity (less clustering, less voids) in the repartition of their points. Using these results, in this paper we study the impact of the $dcx$ ordering of pp on the …
Given two independent Poisson point processes $Φ^{(1)},Φ^{(2)}$ in $R^d$, the continuum AB percolation model is the graph with points of $Φ^{(1)}$ as vertices and with edges between any pair of points for which the intersection of balls of radius $2r$ centred at these points contains at least one point of $Φ^{(2)}$. This is a generalization of the $AB$ percolation model on discrete lattices. We show the existence of percolation for all $d > 1$ and derive bounds for a critical intensity. We also provide a characterization for this critical intensity when $d = 2$. To study the connectivity probl…
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 propose an algorithm for the closed-form recursive computation of joint moments and cumulants of all orders for k-hop counts in the 1D unit disk random graph model with Poisson distributed vertices. Our approach uses decompositions of k-hop counts into multiple Poisson stochastic integrals. As a consequence, using the Stein method we derive Berry-Esseen bounds for the asymptotic convergence of renormalized k-hop path counts to the normal distribution as the density of Poisson vertices tends to infinity. Computer codes for the recursive symbolic computation of moments and cumulants are provi…
Random geometric graphs consist of randomly distributed nodes (points), with pairs of nodes within a given mutual distance linked. In the usual model the distribution of nodes is uniform on a square, and in the limit of infinitely many nodes and shrinking linking range, the number of isolated nodes is Poisson distributed, and the probability of no isolated nodes is equal to the probability the whole graph is connected. Here we examine these properties for several self-similar node distributions, including smooth and fractal, uniform and nonuniform, and finitely ramified or otherwise. We show t…
Consider a dynamic random geometric social network identified by $s_t$ independent points $x_t^1,\ldots,x_t^{s_t}$ in the unit square $[0,1]^2$ that interact in continuous time $t\geq 0$. The generative model of the random points is a Poisson point measures. Each point $x_t^i$ can be active or not in the network with a Bernoulli probability $p$. Each pair being connected by affinity thanks to a step connection function if the interpoint distance $\|x_t^i-x_t^j\|\leq a_\mathsf{f}^\star$ for any $i\neq j$. We prove that when $a_\mathsf{f}^\star=\sqrt{\frac{(s_t)^{l-1}}{pπ}}$ for $l\in(0,1)$, the…
Investigating a model of scale-invariant random spatial network suggested by Aldous, Kendall constructed a random metric $T$ on $\mathbb{R}^d$, for which the distance between points is given by the optimal connection time, when travelling on the road network generated by a Poisson process of lines with a speed limit. In this paper, we look into some fractal properties of that random metric. In particular, although almost surely the metric space $\left(\mathbb{R}^d,T\right)$ is homeomorphic to the usual Euclidean $\mathbb{R}^d$, we prove that its Hausdorff dimension is given by $(γ-1)d/(γ-d)>d$…
Lipschitz cutset for fractal graphs and applications to the spread of infections
该研究将依赖粒子过程的Lipschitz渗流推广到分形图的时空粗粒化框架，证明当恢复速率足够小时，分形图上由单粒子起始的感染能以正概率无限存活。
We study the homology of random Čech complexes generated by a homogeneous Poisson process. We focus on 'homological connectivity' - the stage where the random complex is dense enough, so that its homology "stabilizes" and becomes isomorphic to that of the underlying topological space. Our results form a comprehensive high-dimensional analogue of well-known phenomena related to connectivity in the Erdős-Rényi graph and random geometric graphs. We first prove that there is a sharp phase transition describing homological connectivity. Next, we analyze the behavior of the complex in the critical w…
Let S_{n,k} denote the random geometric graph obtained by placing points inside a square of area n according to a Poisson point process of intensity 1 and joining each such point to the k=k(n) points of the process nearest to it. In this paper we show that if Pr(S_{n,k} connected) > n^{-γ_1} then the probability that S_{n,k} contains a pair of `small' components `close' to each other is o(n^{-c_1}) (in a precise sense of `small' and 'close'), for some absolute constants γ_1>0 and c_1 >0. This answers a question of Walters. (A similar result was independently obtained by Balister.) As an applic…
Recent Progress on Fractal Percolation
本文综述了分形渗流过程的近期研究进展，重点探讨了极限集的维数理论及超临界状态下非平凡连通分量的几何性质，并揭示了其Assouad谱和中间维数恒等于盒维数的结论。
We investigate the random continuous trees called Lévy trees, which are obtained as scaling limits of discrete Galton-Watson trees. We give a mathematically precise definition of these random trees as random variables taking values in the set of equivalence classes of compact rooted R-trees, which is equipped with the Gromov-Hausdorff distance. To construct Lévy trees, we make use of the coding by the height process which was studied in detail in previous work. We then investigate various probabilistic properties of Lévy trees. In particular we establish a branching property analogous to the w…
Graph connectivity with fixed endpoints in the random-connection model
本文研究基于 Poisson 点过程的随机连接模型中固定端点子图计数，提出计算任意阶累积量与矩的组合表达式，推导出中心极限定理与连通性界限，并通过 SageMath 进行数值验证。
We consider the interlacement Poisson point process on the space of doubly-infinite Z^d-valued trajectories modulo time-shift, tending to infinity at positive and negative infinite times. The set of vertices and edges visited by at least one of these trajectories is the random interlacement at level u of Sznitman arXiv:0704.2560 . We prove that for any u>0, almost surely, (1) any two vertices in the random interlacement at level u are connected via at most ceiling(d/2) trajectories of the point process, and (2) there are vertices in the random interlacement at level u which can only be connect…
Line-of-sight Cox percolation on Poisson-Delaunay triangulation
本文研究了城市环境下D2D网络的渗流特性，通过Poisson-Delaunay三角剖分建模街道系统，利用Cox和Bernoulli过程描述用户分布，并引入增强技术首次在通信网络中应用渗流工具，分析了连通图的渗流状态。
Maximum degree for random geometric graphs in the connectivity regime and beyond
本文研究随机几何图达到目标最大度所需阈值半径的渐近波动，在目标度增长阶介于log n与(log n)^d之间时，证明波动服从耿贝尔分布，且高度顶点的度标记点过程依分布收敛于泊松点过程。
Let $G=G_{n,k}$ denote the graph formed by placing points in a square of area $n$ according to a Poisson process of density 1 and joining each point to its $k$ nearest neighbours. Balister, Bollobás, Sarkar and Walters proved that if $k<0.3043\log n$ then the probability that $G$ is connected tends to 0, whereas if $k>0.5139\log n$ then the probability that $G$ is connected tends to 1. We prove that, around the threshold for connectivity, all vertices near the boundary of the square are part of the (unique) giant component. This shows that arguments about the connectivity of $G$ do not need to…
In this paper we study the homology of a random Cech complex generated by a homogeneous Poisson process in a compact Riemannian manifold M. In particular, we focus on the phase transition for "homological connectivity" where the homology of the complex becomes isomorphic to that of M. The results presented in this paper are an important generalization of [7], from the flat torus to general compact Riemannian manifolds. In addition to proving the statements related to homological connectivity, the methods we develop in this paper can be used as a framework for translating results for random geo…
We define a class of random measures, spatially independent martingales, which we view as a natural generalisation of the canonical random discrete set, and which includes as special cases many variants of fractal percolation and Poissonian cut-outs. We pair the random measures with deterministic families of parametrised measures $\{η_t\}_t$, and show that under some natural checkable conditions, a.s. the total measure of the intersections is Hölder continuous as a function of $t$. This continuity phenomenon turns out to underpin a large amount of geometric information about these measures, al…
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