Random walks on decorated Galton-Watson trees

TL;DR

This paper investigates the behavior of simple random walks on complex decorated Galton-Watson trees, analyzing their fractal and spectral properties based on the growth of attached graphs and offspring distribution tail decay.

Contribution

It introduces a model of decorated Galton-Watson trees with detailed analysis of random walk dimensions and fluctuations, extending understanding of random walks on fractal-like structures.

Findings

Determined fractal, spectral, and walk dimensions as functions of model parameters.

Established bounds on fluctuations of these dimensions.

Provided insights into random walk displacement behavior on complex 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.