Mandelbrot Cascades and Related Topics

Barral, Julien

doi:10.1007/978-3-662-43920-3_1

Cited by 4 publications

(3 citation statements)

References 78 publications

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“…Recursive distributional equations of the form (1.3) have been extensively studied, and much is known about the law of random variables satisfying this type of stability and self-similarity; see, e.g., [6,[23][24][25]27] and references therein. In particular, the left and right tail asymptotic of the law μ defined by (1.2) and (1.3) can be obtained as a special case of these results.…”

Section: Introductionmentioning

confidence: 99%

Stationary distribution and cover time of sparse directed configuration models

Caputo

Quattropani

2020

Probab. Theory Relat. Fields

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We consider sparse digraphs generated by the configuration model with given in-degree and out-degree sequences. We establish that with high probability the cover time is linear up to a poly-logarithmic correction. For a large class of degree sequences we determine the exponent $$\gamma \ge 1$$ γ ≥ 1 of the logarithm and show that the cover time grows as $$n\log ^{\gamma }(n)$$ n log γ ( n ) , where n is the number of vertices. The results are obtained by analysing the extremal values of the stationary distribution of the digraph. In particular, we show that the stationary distribution $$\pi $$ π is uniform up to a poly-logarithmic factor, and that for a large class of degree sequences the minimal values of $$\pi $$ π have the form $$\frac{1}{n}\log ^{1-\gamma }(n)$$ 1 n log 1 - γ ( n ) , while the maximal values of $$\pi $$ π behave as $$\frac{1}{n}\log ^{1-\kappa }(n)$$ 1 n log 1 - κ ( n ) for some other exponent $$\kappa \in [0,1]$$ κ ∈ [ 0 , 1 ] . In passing, we prove tight bounds on the diameter of the digraphs and show that the latter coincides with the typical distance between two vertices.

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Section: Introductionmentioning

confidence: 99%

Stationary distribution and cover time of sparse directed configuration models

Caputo

Quattropani

2020

Probab. Theory Relat. Fields

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“…Such self-similar variables appear notably in connections with Mandelbrot's multiplicative cascades and branching random walks. A more general version of (9) has been studied by Rösler [37], Liu [29,28,30,31] and Barral [4,5]. Among others, these references provide detailed results concerning the uniqueness of the solution Z 1 , its left and right tails, its positive and negative moments, its support, and even its absolute continuity w.r.t.…”

Section: Introductionmentioning

confidence: 99%

Random walk on sparse random digraphs

Bordenave

Caputo

Salez

2017

Probab. Theory Relat. Fields

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A finite ergodic Markov chain exhibits cutoff if its distance to equilibrium remains close to its initial value over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Originally discovered in the context of card shuffling (Aldous-Diaconis, 1986), this remarkable phenomenon is now rigorously established for many reversible chains.Here we consider the non-reversible case of random walks on sparse directed graphs, for which even the equilibrium measure is far from being understood. We work under the configuration model, allowing both the in-degrees and the out-degrees to be freely specified. We establish the cutoff phenomenon, determine its precise window and prove that the cutoff profile approaches a universal shape. We also provide a detailed description of the equilibrium measure. by n under the equilibrium measure (right) for the random walk on a random digraph with n = 5000 + 5000 + 5000 vertices with respective degrees (d + , d − ) = (3, 2), (3, 4) and (4, 4).

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“…Statistical mechanics and multifractals are well known to be closely related. Typical situations are provided by the energy model associated with a Gibbs measure on the boundary Σ of the dyadic tree Σ * in the context of the thermodynamic formalism [11,29,30], or the random energy model associated with a branching random walk on Σ * , namely directed polymers on disordered trees [1,2,10,13,22,27]. The purpose of this paper is to investigate the thermodynamic and geometric impact of a random sparse sampling on such structures.…”

Section: Introductionmentioning

confidence: 99%

Random Sparse Sampling in a Gibbs Weighted Tree and Phase Transitions

Barral

Seuret²

2018

J. Inst. Math. Jussieu

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Let µ be the geometric realization on [0, 1] of a Gibbs measure on Σ = {0, 1} N associated with a Hölder potential. The thermodynamic and multifractal properties of µ are well known to be linked via the multifractal formalism. In this article, the impact of a random sampling procedure on this structure is studied.More precisely, let {Iw}w∈Σ * stand for the collection of dyadic subintervals of [0, 1] naturally indexed by the set of finite dyadic words Σ * . Fix η ∈ (0, 1), and a sequence (pw)w∈Σ * of independent Bernoulli variables of parameters 2 −|w|(1−η) (|w| is the length of w). We consider the (very sparse) remaining values µ = {µ(Iw) : w ∈ Σ * , pw = 1}.We prove that when η < 1/2, it is possible to entirely reconstruct µ from the sole knowledge of µ, while it is not possible when η > 1/2, hence a first phase transition phenomenon.We show that, for all η ∈ (0, 1), it is possible to reconstruct a large part of the initial multifractal structure of µ, via the fine study of µ. After reorganization, these coefficients give rise to a random capacity with new remarkable scaling and multifractal properties: its L q -spectrum exhibits two phase transitions, and has a rich thermodynamic and geometric structure.• Recovering from sparse information: The set of surviving vertices S j (η) has a cardinality of expectation 2 djη (which is exponentially less than the 2 dj initial coefficients), and is very sparse. The first question concerns the information remaining after the sampling operation. Can one recover the initial Gibbs measure µ (i.e. all the values (µ(I w )) w∈Σ * ) from the sole knowledge of µ?If not, what about recovering the free energy and multifractal spectrum of µ? • Structure of µ: The new object µ is not a capacity any more. Does it have a well-organized structure though? The last two above questions are of course related to each other. Concerning the reconstruction problematics, recovering the scaling behavior from sparse information is a very natural issue in signal processing (this is one issue in compressive sensing). This allows one to evaluate the "incompressible" information represented by the initial capacity. We bring an answer when µ is the geometric realization on [0, 1] of a Gibbs measure associated with a Hölder continuous potential on Σ, and more generally a non trivial Gibbs capacity. Specifically, the capacity µ satisfies that there exists a Gibbs measure ν, K > 0 and (α, β) ∈ R + × R + such thatand µ is not constant, so (α, β) = (0, 0) (see Section 2.1 for a precise definition of Gibbs measures and capacities).Theorem 1. Suppose that µ is a Gibbs capacity. With probability one, when η < 1/2, one can reconstruct, up to some multiplicative constant depending only on µ, all the values {µ(I w ) : w ∈ Σ * }, while when η > 1/2, it is impossible provided µ is not built from a potential on Σ depending on only finitely many letters.See Section 3, Theorem 4 for a more precise statement. This constitutes a first phase transition phenomenon at η = 1/2.Regarding recovering of statistical and geometri...

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Mandelbrot Cascades and Related Topics

Cited by 4 publications

References 78 publications

Stationary distribution and cover time of sparse directed configuration models

Stationary distribution and cover time of sparse directed configuration models

Random walk on sparse random digraphs

Random Sparse Sampling in a Gibbs Weighted Tree and Phase Transitions

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