The developing theory of geometric random walks is outlined here. Three aspects — general methods for estimating convergence (the “mixing” rate), isoperimetric inequalities in Rn and their intimate connection to random walks, and algorithms for fundamental problems (volume computation and convex optimization) that are based on sampling by random walks — are discussed.

We present a systematic study of the expected complexity of the intersection of geometric objects. We first study the expected size of the intersection between a random Voronoi diagram and a generic geometric object that consists of a finite collection of line segments in the plane. Using this result, we explore the intersection complexity of a random Voronoi diagram with the following target o...

In this paper we study the synchronization properties of random geometric graphs. We show that the onset of synchronization takes place roughly at the same value of the order parameter that a random graph with the same size and average connectivity. However, the dependence of the order parameter with the coupling strength indicates that the fully synchronized state is more easily attained in ra...

Let P be a Poisson process of intensity one in a square Sn of area n. We construct a random geometric graph Gn,k by joining each point of P to its k nearest neighbours. For many applications it is desirable that Gn,k is highly connected, that is, it remains connected even after the removal of a small number of its vertices. In this paper we relate the study of the s-connectivity of Gn,k to our ...

Operator geometric stable laws are the weak limits of operator normed and centered geometric random sums of independent, identically distributed random vectors. They generalize operator stable laws and geometric stable laws. In this work we characterize operator geometric stable distributions, their divisibility and domains of attraction, and present their application to finance. Operator geome...

The complexity of geometric algorithms is often express in terms of the input size and sometimes of the output size, but the behavior of these algorithms may depend drastically on the geometric distribution of the input. The difference between the best-case and the worst-case can be important. An alternative is to analyze these algorithms under the hypothesis of a probabilistic distribution of ...

Tail distribution bounds play a major role in the estimation of failure probabilities in performance and reliability analysis of systems. They are usually estimated using the Markov and Chebyshev’s inequalities, which represent tail distribution bounds for a random variable in terms of its mean or variance. This paper presents the formal verification of Markov’s and Chebyshev’s inequalities for...