Rescaled weighted random balls models and stable self-similar random fields

Functional macroscopic behavior of weighted random ball model

We consider a generalization of the weighted random ball model. The model is driven by a random Poisson measure with a product heavy tailed intensity measure. Such a model typically represents the transmission of a network of stations with a fading effect. In a previous article, the authors proved the convergence of the finite-dimensional distributions of related generalized random fields under various scalings and in the particular case when the fading function is the indicator function of the unit ball. In this paper, tightness and functional convergence are investigated. Using suitable mome…

Intrinsic area near the origin for self-similar growth-fragmentations and related random surfaces

We study the behaviour of a natural measure defined on the leaves of the genealogical tree of some branching processes, namely self-similar growth-fragmentation processes. Each particle, or cell, is attributed a positive mass that evolves in continuous time according to a positive self-similar Markov process and gives birth to children at negative jumps events. We are interested in the asymptotics of the mass of the ball centered at the root, as its radius decreases to $0$. We obtain the almost sure behaviour of this mass when the Eve cell starts with a strictly positive size. This differs fro…

Kozma 的中心激发游走收敛到欧几里得球

Kozma's centrally excited walk converges to a Euclidean ball

本文证明了 Kozma 关于中心激发随机游走在所有维度 $d\geq2$ 中访问集收敛到欧几里得球的猜想，并给出访问次数均匀收敛到锥体的结果，同时推广到一般范数情形。

Scaling limits for simple random walks on random ordered graph trees

Consider a family of random ordered graph trees $(T_n)_{n\geq 1}$, where $T_n$ has $n$ vertices. It has previously been established that if the associated search-depth processes converge to the normalised Brownian excursion when rescaled appropriately as $n\rightarrow\infty$, then the simple random walks on the graph trees have the Brownian motion on the Brownian continuum random tree as their scaling limit. Here, this result is extended to demonstrate the existence of a diffusion scaling limit whenever the volume measure on the limiting real tree is nonatomic, supported on the leaves of the l…

Macroscopic analysis of shot-noise Cox random balls

In this paper, we consider a cluster model of weighted Euclidean random balls generated by a shot-noise Cox process. It is an example of cluster point process. We perform a scaling on the model by shrinking the radii of the balls and compensate this effect by increasing the (mean) number of balls in each cluster, or/and increasing the (mean) number of clusters. We consider two different scenarios, say a local and a global scenarios. Heuristically, in the first scenario, we focus on the mean number of large balls in a cluster while in the second one, we focus on the global mean number of large …

Random Balls Model With Dependence

In this article, we consider a configuration of weighted random balls in $\mathbb{R}^d$ generated according to a Poisson point process. The model investigated exhibits inhomogeneity, as well as dependence between the centers and the radii and heavy tails phenomena. We investigate the asymptotic behavior of the total mass of the configuration of the balls at a macroscopic level. Three different regimes appear depending on the intensity parameters and the zooming factor. Among the three limiting fields, two are stable while the third one is a Poisson integral bridging between the two stable regi…

随机内接多面体逼近的Hausdorff逼近极限定理

A limit theorem for Hausdorff approximation by random inscribed polytopes

本文研究用边界随机采样点的凸包逼近光滑凸体，证明在最优密度下Hausdorff逼近误差经重标度后趋于Gumbel分布。

关于加权硬核模型与全等欧氏球的拉多覆盖问题

On the weighted hard-core model and Rado's covering problem for congruent Euclidean balls

本文针对全等欧氏球的覆盖问题，通过纯组合论证和加权硬核模型方法，改进了经典维塔利覆盖引理给出的体积下界，获得了更优的数学结果。