Random walks on decorated Galton-Watson trees

Abstract

In this article, we study a simple random walk on a decorated Galton-Watson tree, obtained from a Galton-Watson tree by replacing each vertex of degree $n$ with an independent copy of a graph $G_n$ and gluing the inserted graphs along the tree structure. We assume that there exist constants $d, R \geq 1, v < \infty$ such that the diameter, effective resistance across and volume of $G_n$ respectively grow like $n^{\frac{1}{d}}, n^{\frac{1}{R}}, n^v$ as $n \to \infty$. We also assume that the underlying Galton-Watson tree is critical with offspring tails decaying like $cx^{-\alpha}$ for some constant $c>0$ and some $\alpha \in (1,2)$. We establish the fractal dimension, spectral dimension, walk dimension and simple random walk displacement exponent for the resulting metric space as functions of $\alpha, d, R$ and $v$, along with bounds on the fluctuations of these quantities.