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RANDOM SPARSE SAMPLING IN A GIBBS WEIGHTED TREE AND PHASE TRANSITIONS

Part of: Dynamical systems with hyperbolic behavior Classical measure theory Special processes Limit theorems

Published online by Cambridge University Press:  09 May 2018

Julien Barral  and
Stéphane Seuret
Julien Barral
Affiliation:
LAGA, CNRS UMR 7539, Institut Galilée, Université Paris 13, Sorbonne Paris Cité, 99 avenue Jean-Baptiste Clément, 93430 Villetaneuse, France DMA, CNRS UMR 8553, Ecole Normale Supérieure, 45 rue d’ULM, 75005 Paris, France ([email protected])
Stéphane Seuret
Affiliation:
Université Paris-Est, LAMA (UMR 8050), UPEMLV, UPEC, CNRS, F-94010, Créteil, France ([email protected])
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Abstract

Let $\unicode[STIX]{x1D707}$ be the projection on $[0,1]$ of a Gibbs measure on $\unicode[STIX]{x1D6F4}=\{0,1\}^{\mathbb{N}}$ (or more generally a Gibbs capacity) associated with a Hölder potential. The thermodynamic and multifractal properties of $\unicode[STIX]{x1D707}$ are well known to be linked via the multifractal formalism. We study the impact of a random sampling procedure on this structure. More precisely, let $\{{I_{w}\}}_{w\in \unicode[STIX]{x1D6F4}^{\ast }}$ stand for the collection of dyadic subintervals of $[0,1]$ naturally indexed by the finite dyadic words. Fix $\unicode[STIX]{x1D702}\in (0,1)$, and a sequence $(p_{w})_{w\in \unicode[STIX]{x1D6F4}^{\ast }}$ of independent Bernoulli variables of parameters $2^{-|w|(1-\unicode[STIX]{x1D702})}$. We consider the (very sparse) remaining values $\widetilde{\unicode[STIX]{x1D707}}=\{\unicode[STIX]{x1D707}(I_{w}):w\in \unicode[STIX]{x1D6F4}^{\ast },p_{w}=1\}$. We study the geometric and statistical information associated with $\widetilde{\unicode[STIX]{x1D707}}$, and the relation between $\widetilde{\unicode[STIX]{x1D707}}$ and $\unicode[STIX]{x1D707}$. To do so, we construct a random capacity $\mathsf{M}_{\unicode[STIX]{x1D707}}$ from $\widetilde{\unicode[STIX]{x1D707}}$. This new object fulfills the multifractal formalism, and its free energy is closely related to that of $\unicode[STIX]{x1D707}$. Moreover, the free energy of $\mathsf{M}_{\unicode[STIX]{x1D707}}$ generically exhibits one first order and one second order phase transition, while that of $\unicode[STIX]{x1D707}$ is analytic. The geometry of $\mathsf{M}_{\unicode[STIX]{x1D707}}$ is deeply related to the combination of approximation by dyadic numbers with geometric properties of Gibbs measures. The possibility to reconstruct $\unicode[STIX]{x1D707}$ from $\widetilde{\unicode[STIX]{x1D707}}$ by using the almost multiplicativity of $\unicode[STIX]{x1D707}$ and concatenation of words is discussed as well.

Keywords

Hausdorff dimensionrandom samplingthermodynamic formalismphase transitionsmetric number theoryubiquity theorymultifractalslarge deviations

MSC classification

Primary: 28A78: Hausdorff and packing measures 28A80: Fractals 37D35: Thermodynamic formalism, variational principles, equilibrium states 60F10: Large deviations 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 1 , January 2020 , pp. 65 - 116
Copyright
© Cambridge University Press 2018 

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