Critical Self-Similar Markov Trees

TL;DR

This paper investigates critical self-similar Markov trees, extending previous work by constructing these trees, analyzing their fractal dimensions, and exploring their harmonic and length measures using advanced probabilistic techniques.

Contribution

It introduces the construction of critical self-similar Markov trees and analyzes their fractal and measure-theoretic properties, completing the theoretical framework for these structures.

Findings

Constructed critical self-similar Markov trees

Computed their fractal dimensions

Analyzed harmonic and length measures using spinal decomposition

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 A\"id\'ekon--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.