Charles Bordenave

Around the circular law

These expository notes are centered around the circular law theorem, which states that the empirical spectral distribution of a nxn random matrix with i.i.d. entries of variance 1/n tends to the uniform law on the unit disc of the complex plane as the dimension

Performance of random medium access control, an asymptotic approach

n

The Radial Spanning Tree of a Poisson Point Process

tends to infinity. This phenomenon is the non-Hermitian counterpart of the semi circular limit for Wigner random Hermitian matrices, and the quarter circular limit for Marchenko-Pastur random covariance matrices. We present a proof in a Gaussian case, due to Silverstein, based on a formula by Ginibre, and a proof of the universal case by revisiting the approach of Tao and Vu, based on the Hermitization of Girko, the logarithmic potential, and the control of the small singular values. Beyond the finite variance model, we also consider the case where the entries have heavy tails, by using the objective method of Aldous and Steele borrowed from randomized combinatorial optimization. The limiting law is then no longer the circular law and is related to the Poisson weighted infinite tree. We provide a weak control of the smallest singular value under weak assumptions, using asymptotic geometric analysis tools. We also develop a quaternionic Cauchy-Stieltjes transform borrowed from the Physics literature.

Large Deviations of Poisson Cluster Processes

Random Medium-Access-Control (MAC) algorithms have played an increasingly important role in the development of wired and wireless Local Area Networks (LANs) and yet the performance of even the simplest of these algorithms, such as slotted-Aloha, are still not clearly understood. In this paper we provide a general and accurate method to analyze networks where interfering users share a resource using random MAC algorithms. We show that this method is asymptotically exact when the number of users grows large, and explain why it also provides extremely accurate performance estimates even for small systems. We apply this analysis to solve two open problems: (a) We address the stability region of non-adaptive Aloha-like systems. Specifically, we consider a fixed number of buffered users receiving packets from independent exogenous processes and accessing the resource using Aloha-like algorithms. We provide an explicit expression to approximate the stability region of this system, and prove its accuracy. (b) We outline how to apply the analysis to predict the performance of adaptive MAC algorithms, such as the exponential back-off algorithm, in a system where saturated users interact through interference. In general, our analysis may be used to quantify how far from optimality the simple MAC algorithms used in LANs today are, and to determine if more complicated (e.g. queue-based) algorithms proposed in the literature could provide significant improvement in performance.

The dead leaves model: a general tessellation modeling occlusion

We analyze a class of random spanning trees built on a realization of an homogeneous Poisson point process of the plane. This tree has a local construction rule and a radial structure with the origin as its root We first use stochastic geometry arguments to analyze local functionals of the random tree such as the distribution of the length of the edges or the mean degree of the vertices. Far away from the origin, these local properties are shown to be close to those of the directed spanning tree introduced by Bhatt and Roy. We then use the theory of continuous state space Markov chains to analyze some non local properties of the tree such as the shape and structure of its semi-infinite paths or the shape of the set of its vertices less than

Combinatorial Optimization Over Two Random Point Sets

k

Spectrum of large random reversible Markov chains: Heavy-tailed weights on the complete graph.

generations away from the origin. This class of spanning trees has applications in many fields and in particular in communications.

The rank of diluted random graphs

In this paper we prove scalar and sample path large deviation principles for a large class of Poisson cluster processes. As a consequence, we provide a large deviation principle for ergodic Hawkes point processes.

Decentralized constraint satisfaction

In this article, we study a particular example of general random tessellation, the dead leaves model. This model, first studied by the mathematical morphology school, is defined as a sequential superimposition of random closed sets, and provides the natural tool to study the occlusion phenomenon, an essential ingredient in the formation of visual images. We generalize certain results of G. Matheron and, in particular, compute the probability of n compact sets being included in visible parts. This result characterizes the distribution of the boundary of the dead leaves tessellation.

Large deviations of empirical neighborhood distribution in sparse random graphs

Let \((\mathcal{X},\mathcal{Y})\) be a pair of random point sets in \({\mathbb{R}}^{d}\) of equal cardinal obtained by sampling independently 2n points from a common probability distribution μ. In this paper, we are interested by functions L of \((\mathcal{X},\mathcal{Y})\) which appear in combinatorial optimization. Typical examples include the minimal length of a matching of \(\mathcal{X}\) and \(\mathcal{Y}\), the length of a traveling salesperson tour constrained to alternate between points of each set, or the minimal length of a connected bipartite r-regular graph with vertex set \((\mathcal{X},\mathcal{Y})\). As the size n of the point sets goes to infinity, we give sufficient conditions on the function L and the probability measure μ which guarantee the convergence of \(L(\mathcal{X},\mathcal{Y})\) under a suitable scaling. In the case of the minimal length matching, we extend results of Dobric and Yukich, and Boutet de Monvel and Martin.