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Critical Self-Similar Markov Trees

Nicolas Curien, Xingjian Hu, Dongjian Qian

Abstract

Recently introduced and studied in arXiv:2407.07888, a self-similar Markov tree (ssMt) is a random decorated tree that vastly generalises the fragmentation tree. We study here the critical case that was left aside in arXiv:2407.07888. Borrowing techniques from branching random walk, in particular the recent result of Aïdékon--Hu--Shi arXiv:2409.01048, we can complete the picture by constructing critical ssMt, computing their fractal dimension and studying their associated harmonic and length measures using spinal decomposition.

Critical Self-Similar Markov Trees

Abstract

Recently introduced and studied in arXiv:2407.07888, a self-similar Markov tree (ssMt) is a random decorated tree that vastly generalises the fragmentation tree. We study here the critical case that was left aside in arXiv:2407.07888. Borrowing techniques from branching random walk, in particular the recent result of Aïdékon--Hu--Shi arXiv:2409.01048, we can complete the picture by constructing critical ssMt, computing their fractal dimension and studying their associated harmonic and length measures using spinal decomposition.
Paper Structure (22 sections, 21 theorems, 172 equations, 2 figures)

This paper contains 22 sections, 21 theorems, 172 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Measures and spine decomposition.
  3. Acknowledgments.
  4. Background and construction of ssMt
  5. Background
  6. Critical self-similar Markov trees
  7. Approximation by subcritical ssMt
  8. Examples
  9. Measures on critical ssMt
  10. Tools from Branching processes
  11. Fluctuation theory for the Lévy process $\hat{\xi}$
  12. Length measure of the tree below a barrier
  13. The harmonic measure
  14. The Hausdorff dimension of the leaves
  15. Properties of critical ssMt
  16. ...and 7 more sections

Key Result

Theorem 1.2

Under Assumption A the construction in BertoinJean2024SMta can indeed be performed and it yields a family of laws $(\mathbb{Q}_x)$ of compact decorated random trees satisfying the Markov and self-similarity property. Furthermore, under Assumption B, for any $x>0$ the Hausdorff dimension of the leave

Figures (2)

  • Figure 1: Illustration of the criticality on the cumulant function.
  • Figure 2: Illustration of a decoration-reproduction process $(X,\eta)$ and $(X^{i,\varepsilon},\eta^{i,\varepsilon})$ ($i=1,2,3$) coupled with it. The dots represent the locations of the atoms of the reproduction. The dashed lines and circles represent the original decoration–reproduction process $(X,\eta)$. In the drift case (1), the atoms of the reproduction are both shifted in time and space. In the killing case (2), the atoms are the same, the decoration-reproduction process is just possibly killed earlier. In the reproduction case (3), the atoms stay at the same position but are multiplied by $\mathrm{e}^{- \varepsilon}$.

Theorems & Definitions (45)

  • Theorem 1.2: Construction of the critical ssMt
  • Lemma 2.1: BertoinJean2024SMta Compactly glueable
  • Proposition 2.3: Existence of critical ssMt
  • Remark 2.4
  • proof
  • Proposition 2.5: Subcritical approximations
  • proof
  • Remark 2.6: Intrinsic approximation
  • Example 2.7: Aïdékon & Da Silva aidekon2022growth
  • Example 2.8: Branching Bessel processes
  • ...and 35 more