[Paper Review] Uniform multifractal structure of stable trees

Paul Balança|arXiv (Cornell University)|Aug 2, 2015

Theoretical and Computational Physics33 references3 citations

TL;DR

This paper investigates the multifractal structure of stable trees with index γ ∈ (1,2), focusing on the scaling exponents of two natural measures: local time ℓᵃ and mass measure m. It establishes a uniform multifractal spectrum across all levels a > 0, characterizing the scaling index h ∈ [(1+γ)/γ, γ/(γ−1)] for the mass measure, and links the largest masses on subsets to the packing dimension of the underlying set F, providing sharp bounds on the order of the maximal mass via geometric and probabilistic techniques.

ABSTRACT

In this work, we investigate the spectrum of singularities of random stable trees with parameter $γ\in(1,2)$. We consider for that purpose the scaling exponents derived from two natural measures on stable trees: the local time $\ell^a$ and the mass measure $ extbf{m}$, providing as well a purely geometrical interpretation of the latter exponent. We first characterise the uniform component of the multifractal spectrum which exists at every level $a>0$ of stable trees and corresponds to large masses with scaling index $h\in[ frac{1+γ}γ, fracγ{γ-1}]$ for the mass measure (or equivalently $h\in [ frac{1}γ, frac{1}{γ-1}]$ for the local time). In addition, we investigate the distribution of vertices appearing at random levels with exceptionally large masses of index $h\in[0, frac{1+γ}γ)$. Finally, we discuss more precisely the order of the largest mass existing on any subset $\mathcal{T}(F)$ of a stable tree, characterising the former with the packing dimension of the set $F$.

Motivation & Objective

• To characterize the uniform component of the multifractal spectrum of stable trees across all levels a > 0.
• To investigate the distribution of vertices with exceptionally large masses at scaling indices h ∈ [0, (1+γ)/γ).
• To determine the order of the largest mass on subsets 𝒯(F) of a stable tree in relation to the packing dimension of F.
• To provide a geometric interpretation of the mass measure scaling exponent and relate it to the local time exponent via a transformation.

Proposed method

• Uses the local time measure ℓᵃ on level sets 𝒯(a) to define a scaling exponent for the population mass at height a.
• Defines the mass measure m(dσ) = ∫₀^∞ ℓᵃ(dσ) da, which is diffuse and invariant under re-rooting.
• Applies the Borel–Cantelli lemma to analyze the density of vertices with high scaling exponents in subsets of the tree.
• Relies on the work of Joyce and Preiss on packing dimension and upper box dimension to construct sets with prescribed dimension properties.
• Uses the Ray–Knight theorem to connect local time to the law of continuous-state branching processes (CSBPs).
• Applies geometric measure theory tools, including packing dimension and dimension of level sets, to derive bounds on the maximal mass.

Experimental results

Research questions

• RQ1 What is the uniform component of the multifractal spectrum of stable trees for γ ∈ (1,2), and over which range of scaling exponents does it hold?
• RQ2 How are vertices with exceptionally large masses—specifically those with scaling index h ∈ [0, (1+γ)/γ)—distributed across the tree?
• RQ3 What is the precise relationship between the packing dimension of a set F ⊂ (0,∞) and the order of the largest mass on the subset 𝒯(F) of the stable tree?
• RQ4 How do the scaling exponents of the local time measure ℓᵃ and the mass measure m relate, and what is the geometric interpretation of the mass measure exponent?
• RQ5 Under what conditions is the infimum of the scaling exponent realized on 𝒯(F), and how does this relate to the packing dimension of F?

Key findings

• The uniform multifractal spectrum exists for all a > 0 and corresponds to scaling exponents h ∈ [(1+γ)/γ, γ/(γ−1)] for the mass measure m.
• For the local time measure, the corresponding scaling exponent range is h ∈ [1/γ, 1/(γ−1)].
• Vertices with scaling index h ∈ [0, (1+γ)/γ) are distributed such that their existence is guaranteed in any open neighborhood of 𝒯(F) when F has positive packing measure.
• The infimum of the local time scaling exponent over 𝒯(F) is bounded above by (1 − dimₚ F|𝒯)/γ, with equality realized under positive packing measure conditions.
• The same bound applies to the mass measure scaling exponent, with the infimum realized at h = (1 − dimₚ F|𝒯)/γ + 1.
• The maximal mass on any subset 𝒯(F) is characterized by the packing dimension of F, with the order of the largest mass being determined by dimₚ F|𝒯.