Stochastic Geometry and Point Processes Publications

Topics

Dimensions of unimodular random discrete spaces

This work by F.Baccelli, M.-O. Haji-Mirsadeghi, and A. Khezeli, is focused on large scale properties of infinite graphs and discrete subsets of the Euclidean space. It presents two new notions of dimension, namely the unimodular Minkowski and Hausdorff dimensions, which are inspired by the classical Minkowski and Hausdorff dimensions. These dimensions are defined for unimodular discrete spaces, which are defined in this work as a class of random discrete metric spaces with a distinguished point called the origin. These spaces provide a common generalization to stationary point processes under their Palm version and unimodular random rooted graphs.
The main novelty is the use of unimodularity in the definitions where it suggests replacing the infinite sums pertaining to coverings by large balls by the expectation of certain random variables at the origin. In addition, the main manifestation of unimodularity, that is the mass transport principle, is the key element in the proofs and dimension evaluations.
These dimensions are connected to the growth rate of balls. In particular, versions of the mass distribution principle, Billingsley’s lemma, and Frostman’s lemma are established for unimodular discrete spaces.
The dimensions in question are explicitly evaluated or conjectured for a comprehensive set of examples pertaining to the theory of point processes, unimodular random graphs, and self-similarity.

On the Dimension of Unimodular Discrete Spaces, Part I: Definitions and Basic Properties
On the Dimension of Unimodular Discrete Spaces, Part II: Relations with Growth Rate

Publications

2018-01-14_1001080

Poisson Cox Point Processes for Vehicular Networks

Chang-Sik Choi and François Baccelli arXiv preprint arXiv:1801.04556
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2017-06-01_1000994

Spatial Birth Death Wireless Networks

Abishek Sankararaman and François Baccelli IEEE Transactions on Information Theory 63(6): 3964-3982 (2017)
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2017-05-09_1000975 2016-05-20_1000948

Shape Theorems For Poisson Hail on a Bivariate Ground

Francois Baccelli, Hector A. Chang-Lara, and Sergey Foss ArXiv 2016
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2016-01-14_1000623

Point-Shift Foliation of a Point Process

Francois Baccelli and Mir-Omid Haji-Mirsadeghi ArXiv 2016
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2015-05-28_1000151

On Scaling Limits of Power Law Shot-noise Fields

François Baccelli and Anup Biswas To appear in Stochastic Models 2015
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2014-08-06_1000286

The Boolean Model in the Shannon Regime: Three Thresholds and Related Asymptotics

Venkat Anantharam and François Baccelli Arxiv 2014 To appear in Advances in Applied Probability
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2014-07-19_1000288

Compactification of the Action of a Point-Map on the Palm Probability of a Point Process

François Baccelli and Mir-Omid Haji-Mirsadeghi Arxiv 2014 To appear in the Annals of Probability
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2014-04-08_1000146

Mutual Service Processes in R^d, Existence and Ergodicity

François Baccelli, Fabien Mathieu and Ilkka Norros ArXiv 2014
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2000-01-01_1000992

Community Detection on Euclidean Random Graphs

Abishek Sankararaman and François Baccelli ArXiv 2017
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