Search arXivSearch

arXiv · 2202.02509

Asymptotic Critical Radii in Random Geometric Graphs over 3-Dimensional Convex regions

Abstract

This article presents the precise asymptotical distribution of two types of critical transmission radii, defined in terms of k-connectivity and the minimum vertex degree, for a random geometry graph distributed over a 3-Dimensional Convex region.

Explore related subjects

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jie Ding, Xiaohua Xu, Shuai Ma, Xinshan Zhu. 2022-02-05. Asymptotic Critical Radii in Random Geometric Graphs over 3-Dimensional Convex regions. https://arxiv.org/abs/2202.02509

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

KEEP EXPLORING

Related papers

Wasserstein Convergence Rate for Empirical Measures of Markov Processes

The convergence rate in Wasserstein distance is estimated for empirical measures of ergodic Markov processes, and the estimate can be sharp in some specific situations. The main result is applied to subordinations of typical models excluded by existing results, which include: stochastic Hamiltonian systems on $\mathbb R^{n}\times \mathbb R^{m}$, spherical velocity Langevin processes on $\mathbb R^n\times\mathbb S^{n-1},$ multi-dimensional Wright-Fisher type diffusion processes, and stable type jump processes.

math.PR

On the Forgetting of Particle Filters

We study the forgetting properties of the particle filter when its state - the collection of particles - is regarded as a Markov chain. Under a strong mixing assumption on the particle filter's underlying Feynman-Kac model, we find that the particle filter is exponentially mixing, and forgets its initial state in $O(\log N )$ 'time', where $N$ is the number of particles and time refers to the number of particle filter algorithm steps, each comprising a selection (or resampling) and mutation (or prediction) operation. We present an example which shows that this rate is optimal. In contrast to our result, available results to date are extremely conservative, suggesting $O(α^N)$ time steps are needed, for some $α>1$, for the particle filter to forget its initialisation. We also study the conditional particle filter (CPF) and extend our forgetting result to this context. We establish a similar conclusion, namely, CPF is exponentially mixing and forgets its initial state in $O(\log N)$ time. To support this analysis, we establish new time-uniform $L^p$ error estimates for CPF, which can be of independent interest. We also establish new propagation-of-chaos type results using our proof techniques, discuss implications to couplings of particle filters and an application to processing out-of-sequence measurements.

math.PR

Universality of the Brownian net

The Brownian web is a collection of one-dimensional coalescing Brownian motions starting from every point in space and time, while the Brownian net is an extension that also allows branching. We show here that the Brownian net is the universal scaling limit of one-dimensional branching-coalescing random walks with weak binary branching and arbitrary increment distributions that have finite $(3+\varepsilon)$-th moment. This gives the first example in the domain of attraction of the Brownian net where paths can cross without coalescing, which poses fundamental technical challenges not present in the non-crossing case.

math.PR