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On the Dimension of Unimodular Discrete Spaces, Part I: Definitions and Basic Properties

François Baccelli 1 Mir-Omid Haji-Mirsadeghi 2 Ali Khezeli 3
1 DYOGENE - Dynamics of Geometric Networks
DI-ENS - Département d'informatique de l'École normale supérieure, CNRS - Centre National de la Recherche Scientifique : UMR 8548, Inria de Paris
Abstract : This work 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. The work is structured in three companion papers which are called Part I-III. Part I (the current paper) introduces unimodular discrete spaces, the new notions of dimensions, and some of their basic properties. Part II is focused on the connection between these dimensions and the growth rate of balls. In particular, it gives versions of the mass distribution principle, Billingsley's lemma, and Frostman's lemma for unimodular discrete spaces. Part III establishes connections with other notions of dimension. It also discusses ergodicity and the non-ergodic cases in more detail than the first two parts. Each part contains a comprehensive set of examples pertaining to the theory of point processes, unimodular random graphs, and self-similarity, where the dimensions in question are explicitly evaluated or conjectured.
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Contributor : Francois Baccelli Connect in order to contact the contributor
Submitted on : Wednesday, January 9, 2019 - 8:34:37 PM
Last modification on : Friday, October 15, 2021 - 1:41:17 PM

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  • HAL Id : hal-01976265, version 1
  • ARXIV : 1807.02980




François Baccelli, Mir-Omid Haji-Mirsadeghi, Ali Khezeli. On the Dimension of Unimodular Discrete Spaces, Part I: Definitions and Basic Properties. 2019. ⟨hal-01976265⟩



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