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On the Dimension of Unimodular Discrete Spaces, Part II: Relations with Growth Rate

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 : The notions of unimodular Minkowski and Hausdorff dimensions are defined in [5] for unimodular random discrete metric spaces. The present paper is focused on the connections between these notions and the polynomial growth rate of the underlying space. It is shown that bounding the dimension is closely related to finding suitable equivariant weight functions (i.e., measures) on the underlying discrete space. The main results are unimodular versions of the mass distribution principle and Billingsley's lemma, which allow one to derive upper bounds on the unimodular Hausdorff dimension from the growth rate of suitable equivariant weight functions. Also, a unimodular version of Frostman's lemma is provided, which shows that the upper bound given by the unimodular Billingsley lemma is sharp. These results allow one to compute or bound both types of unimodular dimensions in a large set of examples in the theory of point processes, unimodular random graphs, and self-similarity. Further results of independent interest are also presented, like a version of the max-flow min-cut theorem for unimodular one-ended trees.
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Contributor : Francois Baccelli Connect in order to contact the contributor
Submitted on : Wednesday, January 9, 2019 - 8:24:14 PM
Last modification on : Friday, October 15, 2021 - 1:41:17 PM

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  • HAL Id : hal-01976257, version 1
  • ARXIV : 1808.02551



François Baccelli, Mir-Omid Haji-Mirsadeghi, Ali Khezeli. On the Dimension of Unimodular Discrete Spaces, Part II: Relations with Growth Rate. 2019. ⟨hal-01976257⟩



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