Spectral dimension of trees with a unique infinite spine

TL;DR

This paper develops a method using generating functions to relate the Hausdorff and spectral dimensions of trees with a unique infinite spine, enabling easier computation for certain random tree models.

Contribution

It introduces a new approach linking dimensions of trees with a spine to generating functions, simplifying analysis for specific models.

Findings

Derived relations between Hausdorff and spectral dimensions.

Applied method to critical non-generic trees.

Obtained new results for the attachment and grafting model.

Abstract

Using generating functions techniques we develop a relation between the Hausdorff and spectral dimension of trees with a unique infinite spine. Furthermore, it is shown that if the outgrowths along the spine are independent and identically distributed, then both the Hausdorff and spectral dimension can easily be determined from the probability generating function of the random variable describing the size of the outgrowths at a given vertex, provided that the probability of the height of the outgrowths exceeding n falls off as the inverse of n. We apply this new method to both critical non-generic trees and the attachment and grafting model, which is a special case of the vertex splitting model, resulting in a simplified proof for the values of the Hausdorff and spectral dimension for the former and novel results for the latter.