Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-21T13:39:08.375Z Has data issue: false hasContentIssue false
  • English
  • Français

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 (barral@math.univ-paris13.fr)
Stéphane Seuret
Affiliation:
Université Paris-Est, LAMA (UMR 8050), UPEMLV, UPEC, CNRS, F-94010, Créteil, France (seuret@u-pec.fr)
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Attia, N. and Barral, J., Hausdorff and packing spectra, large deviations, and free energy for branching random walks in ℝ d , Comm. Math. Phys. 331 (2014), 139–187.Google Scholar
Barral, J., Mandelbrot cascades and related topics, in Geometry and Analysis of Fractals (ed. Feng, D.-J. and Lau, K.-S.), Springer Proceedings in Mathematics and Statistics (Springer, Berlin, Heidelberg, 2014).Google Scholar
Barral, J., Ben Nasr, F. and Peyrière, J., Comparing multifractal formalisms: the neighboring condition, Asian J. Math. 7 (2003), 149–166.Google Scholar
Barral, J. and Seuret, S., Combining multifractal additive and multiplicative chaos, Comm. Math. Phys. 257(2) (2005), 473–497.Google Scholar
Barral, J. and Seuret, S., Heterogeneous ubiquitous systems in ℝ d and Hausdorff dimensions, Bull. Braz. Math. Soc. (N.S.) 38(3) (2007), 467–515.Google Scholar
Barral, J. and Seuret, S., Ubiquity and large intersections properties under digit frequencies constraints, Math. Proc. Cambridge Philos. Soc. 145(3) (2008), 527–548.Google Scholar
Brown, G., Michon, G. and Peyrière, J., On the multifractal analysis of measures, J. Stat. Phys. 66 (1992), 775–790.Google Scholar
Bruin, H. and Leplaideur, R., Renormalization, thermodynamic formalism and quasi-crystals in subshifts, Comm. Math. Phys. 321 (2013), 209–247.Google Scholar
Bruin, H. and Leplaideur, R., Renormalization, freezing phase transitions and Fibonacci quasicristals, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), 739–763, 2015.Google Scholar
Collet, P. and Koukiou, F., Large deviations for multiplicative chaos, Comm. Math. Phys. 147 (1992), 329–342.Google Scholar
Collet, P., Lebowitz, J. L. and Porzio, A., The dimension spectrum of some dynamical systems, J. Stat. Phys. 47 (1987), 609–644.Google Scholar
Dembo, A. and Zeitouni, O., Large Deviations Techniques and Applications (Jones and Bartlett Publishers, Boston, 1993).Google Scholar
Derrida, B. and Spohn, H., Polymers on disordered trees, spin glasses and traveling waves, J. Stat. Phys. 51 (1988), 817–840.Google Scholar
Dodson, M., Melián, M., Pestana, D. and Vélani, S., Patterson measure and Ubiquity, Ann. Acad. Sci. Fenn. Ser. A I Math. 20 (1995), 37–60.Google Scholar
Durand, A., Ubiquitous systems and metric number theory, Adv. Math. 218(2) (2008), 368–394.Google Scholar
Fan, A. H., Feng, D. J. and Wu, J., Recurrence, dimension and entropy, J. Lond. Math. Soc. (2) 64(1) (2001), 229–244.Google Scholar
Feng, D. J., Lyapounov exponents for products of matrices and multifractal analysis. Part I: positive matrices, Israël J. Math. 138 (2003), 353–376.Google Scholar
Feng, D. J. and Lau, K. S., Pressure function for products of non-negative matrices, Math. Res. Lett. 9 (2002), 363–378.Google Scholar
Feng, D.-J. and Olivier, E., Multifractal analysis of the weak Gibbs measures and phase transition- Application to some Bernoulli convolutions, Ergod. Th. & Dynam. Sys. 23 (2003), 1751–1784.Google Scholar
Heurteaux, Y., Estimations de la dimension inférieure et de la dimension supérieure des mesures, Ann. Inst. H. Poincaré Probab. Stat. 34 (1998), 309–338.Google Scholar
Hofbauer, F., Examples for the non uniqueness of the equilibrium state, Trans. Amer. Math. Soc. (1977), 223–241.Google Scholar
Holley, R. and Waymire, E. C., Multifractal dimensions and scaling exponents for strongly bounded random fractals, Ann. Appl. Probab. 2 (1992), 819–845.Google Scholar
Iommi, G. and Todd, M., Transience in dynamical systems, Ergod. Th. & Dynam. Sys. 33 (2013), 1450–1476.Google Scholar
Jaffard, S., On lacunary wavelet series, Ann. Appl. Probab. 10(1) (2000), 313–329.Google Scholar
Jaffard, S., Wavelet techniques in multifractal analysis, in Fractal Geometry and Applications, Proceedings of Symposia in Pure Mathematics, Volume 72, (Part 2) pp. 91–152 (American Mathematical Society, 2004).Google Scholar
Lévy Véhel, J. and Vojak, R., Multifractal analysis of Choquet capacities, Adv. Appl. Math. 20 (1998), 1–43.Google Scholar
Molchan, G. M., Scaling exponents and multifractal dimensions for independent random cascades, Comm. Math. Phys. 179 (1996), 681–702.Google Scholar
Olsen, L., A multifractal formalism, Adv. Math. 116 (1995), 92–195.Google Scholar
Rand, D. A., The singularity spectrum f (𝛼) for cookie-cutters, Ergod. Th. & Dynam. Sys. 9 (1989), 527–541.Google Scholar
Ruelle, D., Thermodynamic formalism, in The Mathematical Structures of Classical Equilibrium Statistical Mechanics, Encyclopedia of Mathematics and its Applications, Vol. 5, (Addison-Wesley Publishing Co., Reading, MA, 1978).Google Scholar
Sarig, O., Phase transitions for countable topological Markov shifts, Comm. Math. Phys. 217 (2001), 555–577.Google Scholar
Schmeling, J., On the completeness of multifractal spectra, Ergod. Th. & Dynam. Sys. 19 (1999), 1595–1616.Google Scholar