Branching Random Walks on Multi-Type Galton–Watson Trees: A Multifractal Analysis
- 1Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia
- 2Department of Mathematics, Faculty of Science, King Khalid University, Abha 61913, Saudi Arabia
Abstract
We study the multifractal structure of a branching random walk on the boundary of a finite-type Galton–Watson tree. For α∈R, we consider the level set EX(α)=t∈∂T:limn→∞SnX(t)n=α. The multi-type structure leads naturally to the weighted mean offspring matrix M(q) and to the deterministic pressure P˜(q)=logρ(M(q)). Using the corresponding Perron–Frobenius eigenvectors, we construct a family of multi-type Mandelbrot martingales and prove their almost sure and Lp uniform convergence on compact subsets of the admissible parameter domain. This yields the associated Mandelbrot measures and the uniform estimates required for the multifractal analysis. As a consequence, for every α in a suitable admissible set I, we obtain almost surely dimHEX(α)=P˜*(α), where P˜* denotes the Legendre transform of P˜ and dimH stands for the Hausdorff dimension. Thus the classical multifractal formalism for one-type branching random walks extends to the genuinely multi-type setting, with the scalar normalization replaced by the Perron–Frobenius spectral structure of the weighted offspring matrices.
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BibTeX
@article{attia2026branching,
title = {{Branching Random Walks on Multi-Type Galton–Watson Trees: A Multifractal Analysis}},
author = {Najmeddine Attia and Mohammed Balegh},
journal = {Fractal and Fractional},
year = {2026},
month = oct,
doi = {10.3390/fractalfract10100731},
url = {https://www.mdpi.com/2504-3110/10/10/731},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Branching Random Walks on Multi-Type Galton–Watson Trees: A Multifractal Analysis
AU - Najmeddine Attia
AU - Mohammed Balegh
PY - 2026
DA - 2026/10/20
JO - Fractal and Fractional
DO - 10.3390/fractalfract10100731
UR - https://www.mdpi.com/2504-3110/10/10/731
AB - We study the multifractal structure of a branching random walk on the boundary of a finite-type Galton–Watson tree. For α∈R, we consider the level set EX(α)=t∈∂T:limn→∞SnX(t)n=α. The multi-type structure leads naturally to the weighted mean offspring matrix M(q) and to the deterministic pressure P˜(q)=logρ(M(q)). Using the corresponding Perron–Frobenius eigenvectors, we construct a family of multi-type Mandelbrot martingales and prove their almost sure and Lp uniform convergence on compact subsets of the admissible parameter domain. This yields the associated Mandelbrot measures and the uniform estimates required for the multifractal analysis. As a consequence, for every α in a suitable admissible set I, we obtain almost surely dimHEX(α)=P˜*(α), where P˜* denotes the Legendre transform of P˜ and dimH stands for the Hausdorff dimension. Thus the classical multifractal formalism for one-type branching random walks extends to the genuinely multi-type setting, with the scalar normalization replaced by the Perron–Frobenius spectral structure of the weighted offspring matrices.
SN - 2504-3110
ER -