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In this paper we study the spectrum of the random geometric graph $G(n,r)$, in a regime where the graph is dense and highly connected. In the \erdren $G(n,p)$ random graph it is well known that upon connectivity the spectrum of the normalized graph Laplacian is concentrated around $1$. We show that such concentration does not occur in the $G(n,r)$ case, even when the graph is dense and almost a complete graph. In particular, we show that the limiting spectral gap is strictly smaller than $1$. In the special case where the vertices are distributed uniformly in the unit cube and $r=1$, we show that for every $0\le k \le d$ there are at least $\binom{d}{k}$ eigenvalues near $1-2^{-k}$, and the limiting spectral gap is exactly $1/2$. We also show that the corresponding eigenfunctions in this case are tightly related to the geometric configuration of the points., Comment: 32 pages, 2 figures

We present normal approximation results at the process level for local functionals defined on dynamic Poisson processes in $\mathbb{R}^d$. The dynamics we study here are those of a Markov birth-death process. We prove functional limit theorems in the so-called thermodynamic regime. Our results are applicable to several functionals of interest in the stochastic geometry literature, including subgraph and component counts in the random geometric graphs.

We introduce Lipschitz-Killing curvature (LKC) regression, a new method to produce $(1-\alpha)$ thresholds for signal detection in random fields that does not require knowledge of the spatial correlation structure. The idea is to fit observed empirical Euler characteristics to the Gaussian kinematic formula via generalized least squares, which quickly and easily provides statistical estimates of the LKCs --- complex topological quantities that can be extremely challenging to compute, both theoretically and numerically. With these estimates, we can then make use of a powerful parametric approximation via Euler characteristics for Gaussian random fields to generate accurate $(1-\alpha)$ thresholds and $p$-values. The main features of our proposed LKC regression method are easy implementation, conceptual simplicity, and facilitated diagnostics, which we demonstrate in a variety of simulations and applications., Comment: 35 pages, 13 figures. 3 tables

We study the homology of simplicial complexes built via deterministic rules from a random set of vertices. In particular, we show that, depending on the randomness that generates the vertices, the homology of these complexes can either become trivial as the number $n$ of vertices grows, or can contain more and more complex structures. The different behaviours are consequences of different underlying distributions for the generation of vertices, and we consider three illustrative examples, when the vertices are sampled from Gaussian, exponential, and power-law distributions in $\R^d$. We also discuss consequences of our results for manifold learning with noisy data, describing the topological phenomena that arise in this scenario as `crackle', in analogy to audio crackle in temporal signal analysis.

For a finite set of points $P$ in $R^d$, the function $d_P: R^d \to R^+$ measures Euclidean distance to the set $P$. We study the number of critical points of $d_P$ when $P$ is a Poisson process. In particular, we study the limit behavior of $N_k$ - the number of critical points of $d_P$ with Morse index $k$ - as the density of points grows. We present explicit computations for the normalized, limiting, expectations and variances of the $N_k$, as well as distributional limit theorems. We link these results to recent results in which the Betti numbers of the random \v{C}ech complex based on $P$ were studied.

For a finite set of points $P$ in $R^d$, the function $d_P: R^d \to R^+$ measures Euclidean distance to the set $P$. We study the number of critical points of $d_P$ when $P$ is a Poisson process. In particular, we study the limit behavior of $N_k$ - the number of critical points of $d_P$ with Morse index $k$ - as the density of points grows. We present explicit computations for the normalized, limiting, expectations and variances of the $N_k$, as well as distributional limit theorems. We link these results to recent results in which the Betti numbers of the random ��ech complex based on $P$ were studied.

This paper considers the problem of detecting equal-shaped non-overlapping unimodal peaks in the presence of Gaussian ergodic stationary noise, where the number, location and heights of the peaks are unknown. A multiple testing approach is proposed in which, after kernel smoothing, the presence of a peak is tested at each observed local maximum. The procedure provides strong control of the family wise error rate and the false discovery rate asymptotically as both the signal-to-noise ratio (SNR) and the search space get large, where the search space may grow exponentially as a function of SNR. Simulations assuming a Gaussian peak shape and a Gaussian autocorrelation function show that desired error levels are achieved for relatively low SNR and are robust to partial peak overlap. Simulations also show that detection power is maximized when the smoothing bandwidth is close to the bandwidth of the signal peaks, akin to the well-known matched filter theorem in signal processing. The procedure is illustrated in an analysis of electrical recordings of neuronal cell activity., Comment: 37 pages, 8 figures

Studying the geometry generated by Gaussian and Gaussian- related random fields via their excursion sets is now a well developed and well understood subject. The purely non-Gaussian scenario has, however, not been studied at all. In this paper we look at three classes of stable random fields, and obtain asymptotic formulae for the mean values of various geometric characteristics of their excursion sets over high levels. While the formulae are asymptotic, they contain enough information to show that not only do stable random fields exhibit geometric behaviour very different from that of Gaussian fields, but they also differ significantly among themselves., 35 pages, 1 figure

We consider vector valued, unit variance Gaussian processes defined over stratified manifolds and the geometry of their excursion sets. In particular, we develop an explicit formula for the expectation of all the Lipschitz--Killing curvatures of these sets. Whereas our motivation is primarily probabilistic, with statistical applications in the background, this formula has also an interpretation as a version of the classic kinematic fundamental formula of integral geometry. All of these aspects are developed in the paper. Particularly novel is the method of proof, which is based on a an approximation to the canonical Gaussian process on the $n$-sphere. The $n\to\infty$ limit, which gives the final result, is handled via recent extensions of the classic Poincar�� limit theorem., Published in at http://dx.doi.org/10.1214/08-AOP439 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)