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Hausdorff and Packing Spectra, Large Deviations, and Free Energy for Branching Random Walks in \({\mathbb{R}^d}\)

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An Erratum to this article was published on 31 January 2017

Abstract

Consider an \({\mathbb{R}^d}\) -valued branching random walk (BRW) on a supercritical Galton Watson tree. Without any assumption on the distribution of this BRW we compute, almost surely and simultaneously, the Hausdorff and packing dimensions of the level sets E(K) of infinite branches in the boundary of the tree (endowed with its standard metric) along which the averages of the BRW have a given closed connected set of limit points K. This goes beyond multifractal analysis, which only considers those level sets when K ranges in the set of singletons \({\{\alpha\}, \alpha \in \mathbb{R}^d}\). We also give a 0–∞ law for the Hausdorff and packing measures of the level sets E({α}), and compute the free energy of the associated logarithmically correlated random energy model in full generality. Moreover, our results complete the previous works on multifractal analysis by including the levels α which do not belong to the range of the gradient of the free energy. This covers in particular a situation that was until now badly understood, namely the case where a first order phase transition occurs. As a consequence of our study, we can also describe the whole singularity spectrum of Mandelbrot measures, as well as the associated free energy function (or L q-spectrum), when a first order phase transition occurs.

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Authors and Affiliations

  1. Département de Mathématiques, Faculté des Sciences de Monastir, 5019, Monastir, Tunisia

    Najmeddine Attia

  2. LAGA (UMR 7539), Département de Mathématiques, Institut Galilée, Université Paris 13, Paris Sorbonne Cité, 99 Avenue Jean-Baptiste Clément, 93430, Villetaneuse, France

    Julien Barral

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  1. Najmeddine Attia
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  2. Julien Barral
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Correspondence to Julien Barral.

Additional information

Communicated by F. Toninelli

The authors were supported by the French National Research Agency Project “DMASC”. They thank the referees for their constructive suggestions, in particular the addition of Theorem 1.4 as a consequence of the main results of the preliminary version. They also thank Dr Xiong Jin for his help in the figures elaboration.

An erratum to this article is available at http://dx.doi.org/10.1007/s00220-016-2826-1.

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Attia, N., Barral, J. Hausdorff and Packing Spectra, Large Deviations, and Free Energy for Branching Random Walks in \({\mathbb{R}^d}\) . Commun. Math. Phys. 331, 139–187 (2014). https://doi.org/10.1007/s00220-014-2087-9

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