The statistical geometry of scale-free random trees

TL;DR

This paper investigates the properties of scale-free random trees, analyzing their geometric and spectral characteristics, including volume distribution, connectivity, and spectral dimensions, to understand their behavior in the thermodynamic limit.

Contribution

It introduces a comprehensive analysis of scale-free random trees, deriving scaling forms and spectral dimensions, advancing understanding of their geometric and spectral properties.

Findings

Derived the scaling form of volume probability

Determined the connectivity dimensions and compared with growth exponents

Calculated the local spectral dimension via Gaussian model analysis

Abstract

The properties of scale-free random trees are investigated using both preconditioning on non-extinction and fixed size averages, in order to study the thermodynamic limit. The scaling form of volume probability is found, the connectivity dimensions are determined and compared with other exponents which describe the growth. The (local) spectral dimension is also determined, through the study of the massless limit of the Gaussian model on such trees.