Limit theory for geometric statistics of point processes having fast decay of correlations

Let $P$ be a simple,stationary point process having fast decay of correlations, i.e., its correlation functions factorize up to an additive error decaying faster than any power of the separation distance. Let $P_n:= P \cap W_n$ be its restriction to windows $W_n:= [-{1 \over 2}n^{1/d},{1 \over 2}n^{1/d}]^d \subset \mathbb{R}^d$. We consider the statistic $H_n^\xi:= \sum_{x \in P_n}\xi(x,P_n)$ where $\xi(x,P_n)$ denotes a score function representing the interaction of $x$ with respect to $P_n$. When $\xi$ depends on local data in the sense that its radius of stabilization has an exponential tail, we establish expectation asymptotics, variance asymptotics, and CLT for $H_n^{\xi}$ and, more generally, for statistics of the re-scaled, possibly signed, $\xi$-weighted point measures $\mu_n^{\xi} := \sum_{x \in P_n} \xi(x,P_n) \delta_{n^{-1/d}x}$, as $W_n \uparrow \mathbb{R}^d$. This gives the limit theory for non-linear geometric statistics (such as clique counts, intrinsic volumes of the Boolean model, and total edge length of the $k$-nearest neighbors graph) of $\alpha$-determinantal point processes having fast decreasing kernels extending the CLTs of Soshnikov (2002) to non-linear statistics. It also gives the limit theory for geometric U-statistics of $\alpha$-permanental point processes and the zero set of Gaussian entire functions, extending the CLTs of Nazarov and Sodin (2012) and Shirai and Takahashi (2003), which are also confined to linear statistics. The proof of the central limit theorem relies on a factorial moment expansion originating in Blaszczyszyn (1995), Blaszczyszyn, Merzbach, Schmidt (1997) to show the fast decay of the correlations of $\xi$-weighted point measures. The latter property is shown to imply a condition equivalent to Brillinger mixing and consequently yields the CLT for $\mu_n^\xi$ via an extension of the cumulant method.Comment: 62 pages. Fundamental changes to the terminology including the title. The earlier 'clustering' condition is now introduced as a notion of mixing and its connection to Brillinger mixing is remarked. Newer results for superposition of independent point processes have been adde

Clustering, percolation and directionally convex ordering of point processes

Heuristics indicate that point processes exhibiting clustering of points have larger critical radius $r_c$ for the percolation of their continuum percolation models than spatially homogeneous point processes. It has already been shown, and we reaffirm it in this paper, that the $dcx$ ordering of point processes is suitable to compare their clustering tendencies. Hence, it was tempting to conjecture that $r_c$ is increasing in $dcx$ order. Some numerical evidences support this conjecture for a special class of point processes, called perturbed lattices, which are "toy models" for determinantal and permanental point processes. However, the conjecture is not true in full generality, since one can construct a Cox point process with degenerate critical radius $r_c=0$, that is $dcx$ larger than a given homogeneous Poisson point process. Nevertheless, we are able to compare some nonstandard critical radii related, respectively, to the finiteness of the expected number of void circuits around the origin and asymptotic of the expected number of long occupied paths from the origin in suitable discrete approximations of the continuum model. These new critical radii sandwich the "true" one. Surprisingly, the inequalities for them go in opposite directions, which gives uniform lower and upper bounds on $r_c$ for all processes $dcx$ smaller than some given process. In fact, the above results hold under weaker assumptions on the ordering of void probabilities or factorial moment measures only. Examples of point processes comparable to Poisson processes in this weaker sense include determinantal and permanental processes. More generally, we show that point processes $dcx$ smaller than homogeneous Poisson processes exhibit phase transitions in certain percolation models based on the level-sets of additive shot-noise fields, as e.g. $k$-percolation and SINR-percolation.Comment: 48 pages, 6 figure

Comparison of the Maximal Spatial Throughput of Aloha and CSMA in Wireless Ad-Hoc Networks

International audienceIn this paper we compare the spatial throughput of Aloha and Carrier Sense Multiple Access (CSMA) in Wireless multihop Ad-Hoc Networks. In other words we evaluate the gain offered by carrier sensing (CSMA) over the pure statiscal collision avoidance which is the basis of Aloha. We use a Signal-to-Interference-and-Noise Ratio (SINR) model where a transmission is assumed to be successful when the SINR is larger than a given threshold. Regarding channel conditions, we consider both standard Rayleigh and negligible fading. For slotted and non-slotted Aloha, we use analytical models as well as simulations to study the density of successful transmissions in the network. As it is very difficult to build precise models for CSMA, we use only simulations to compute the performances of this protocol. We compare the two Aloha versions and CSMA on a fair basis, i.e. when they are optimized to maximize the density of successful transmissions. For slotted Aloha, the key optimization parameter is the medium access probability, for non-slotted Aloha we tune the mean back-off time, whereas for CSMA it is the carrier sense threshold that is adjusted. Our study shows that CSMA always outperforms slotted Aloha, which in turn outperforms its non-slotted version

Moment Closure - A Brief Review

Moment closure methods appear in myriad scientific disciplines in the modelling of complex systems. The goal is to achieve a closed form of a large, usually even infinite, set of coupled differential (or difference) equations. Each equation describes the evolution of one "moment", a suitable coarse-grained quantity computable from the full state space. If the system is too large for analytical and/or numerical methods, then one aims to reduce it by finding a moment closure relation expressing "higher-order moments" in terms of "lower-order moments". In this brief review, we focus on highlighting how moment closure methods occur in different contexts. We also conjecture via a geometric explanation why it has been difficult to rigorously justify many moment closure approximations although they work very well in practice.Comment: short survey paper (max 20 pages) for a broad audience in mathematics, physics, chemistry and quantitative biolog