On the Multifractal Analysis of Branching Random Walk on Galton–Watson Tree with Random Metric

We consider a branching random walk SnX(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_nX(t)$$\end{document} on a supercritical random Galton–Watson tree. We compute the Hausdorff and packing dimensions of the level set E(α)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E(\alpha )$$\end{document} of infinite branches in the boundary of tree endowed with random metric along which the average of SnX(t)/n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_n X(t)/n$$\end{document} have a given limit point.