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We study two branching random walks \(S_n\mathsf {X}(t)\) and \(S_n\mathsf {Y}(t)\) on a supercritical Galton–Watson tree. We determine the Hausdorff and packing dimensions of the set \(\mathrm {X}_\zeta (\alpha ,\beta )\) of infinite branches in the boundary of the tree, endowed with a random metric, for which the ratio \(S_n\mathsf {X}(t)/S_n\mathsf {Y}(t)\) converges to \(\alpha \) while \(S_n\mathsf {Y}(t)/n\) converges to \(\beta \) .

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Relative Multifractal Formalism of Branching Random Walk with Random Metric

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  • 发表日期：

    2026年

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  • Najmeddine Attia,
  • Amal Mahjoub

摘要

We study two branching random walks \(S_n\mathsf {X}(t)\) and \(S_n\mathsf {Y}(t)\) on a supercritical Galton–Watson tree. We determine the Hausdorff and packing dimensions of the set \(\mathrm {X}_\zeta (\alpha ,\beta )\) of infinite branches in the boundary of the tree, endowed with a random metric, for which the ratio \(S_n\mathsf {X}(t)/S_n\mathsf {Y}(t)\) converges to \(\alpha \) while \(S_n\mathsf {Y}(t)/n\) converges to \(\beta \) .