Extremal lifetimes of persistent cycles

Persistent homology captures the appearances and disappearances of topological features such as loops and cavities when growing disks centered at a Poisson point process. We study extreme values for the lifetimes of features dying in bounded components and with birth resp. death time bounded away from the threshold for continuum percolation and the coexistence region. First, we describe the scaling of the minimal lifetimes for general feature dimensions, and of the maximal lifetimes for cavities in the Čech filtration. Then, we proceed to a more refined analysis and establish Poisson approximation for large lifetimes of cavities and for small lifetimes of loops. Finally, we also study the scaling of minimal lifetimes in the Vietoris-Rips setting and point to a surprising difference to the Čech filtration.

Cellular coverage probability is independent of base station density under stochastic geometric models

Stochastic geometry (SG) has been extensively used to model cellular communications, under the assumption that the base stations (BS) are deployed as a Poisson point process in the Euclidean plane. This has spawned a huge number of articles over the past years for different scenarios, culminating in an equally huge number of expressions for the coverage probability in both the uplink (UL) and downink (DL) directions. The trouble is that those expressions include the BS density, $\lambda$, which we prove irrelevant in this article. We start by developing a SG model for a baseline cellular scenario, then prove that the coverage probability is independent of $\lambda$, contrary to popular belief.

On Smooth Mesoscopic Linear Statistics of the Eigenvalues of Random Permutation Matrices

We study the limiting behavior of smooth linear statistics of the spectrum of random permutation matrices in the mesoscopic regime, when the permutation follows one of the Ewens measures on the symmetric group. If we apply a smooth enough test function f to all the determinations of the eigenangles of the permutations, we get a convergence in distribution when the order of the permutation tends to infinity. Two distinct kinds of limit appear: if $$f(0)\ne 0$$ f ( 0 ) ≠ 0 , we have a central limit theorem with a logarithmic variance; and if $$f(0) = 0$$ f ( 0 ) = 0 , the convergence holds without normalization and the limit involves a scale-invariant Poisson point process.

Interference Analysis for mmWave Automotive Radar Considering Blockage Effect

Due to the increasing number of vehicles equipped with millimeter wave (mmWave) radars, interference among automotive radars is becoming a major issue. This paper explores the automotive radar interference in both two-lane and multi-lane scenarios using stochastic geometry. We derive closed-form expressions for mean and variance of interference power considering directional antenna with constant and Gaussian decaying gains. In view of the sensitivity of mmWave radar signals to the blockages, we propose a blockage model including partially and completely blocking, and then calculate the effective number of the interferers. By means of modeling randomness for interferers and blockages as Poisson point process, we characterize the statistics of radar interference under different conditions. We further utilize the interference characterization to estimate the successful ranging probability of automotive radars. These theoretical analyses are verified by using Monte Carlo simulations. The results show that the increasing interfering density and ranging distance largely degrade the radar detection performance, whereas the interference levels decrease as blockage intensity increases.

A numerical method for computing interval distributions for an inhomogeneous Poisson point process modified by random dead times

The inhomogeneous Poisson point process is a common model for time series of discrete, stochastic events. When an event from a point process is detected, it may trigger a random dead time in the detector, during which subsequent events will fail to be detected. It can be difficult or impossible to obtain a closed-form expression for the distribution of intervals between detections, even when the rate function (often referred to as the intensity function) and the dead-time distribution are given. Here, a method is presented to numerically compute the interval distribution expected for any arbitrary inhomogeneous Poisson point process modified by dead times drawn from any arbitrary distribution. In neuroscience, such a point process is used to model trains of neuronal spikes triggered by the detection of excitatory events while the neuron is not refractory. The assumptions of the method are that the process is observed over a finite observation window and that the detector is not in a dead state at the start of the observation window. Simulations are used to verify the method for several example point processes. The method should be useful for modeling and understanding the relationships between the rate functions and interval distributions of the event and detection processes, and how these relationships depend on the dead-time distribution.