The spectral dimension of random trees

TL;DR

This paper introduces a rigorous method to determine the spectral dimension of random trees using the Gaussian model, providing evidence for new scaling hypotheses and relations between spectral and connectivity dimensions.

Contribution

It offers a novel, rigorous approach to calculating spectral dimensions of random trees and supports new hypotheses linking spectral and connectivity dimensions.

Findings

Established a rigorous method for spectral dimension calculation.

Provided evidence for a new scaling hypothesis for Gaussian models.

Supported a conjectured relation between spectral and connectivity dimensions.

Abstract

We present a simple yet rigorous approach to the determination of the spectral dimension of random trees, based on the study of the massless limit of the Gaussian model on such trees. As a byproduct, we obtain evidence in favor of a new scaling hypothesis for the Gaussian model on generic bounded graphs and in favor of a previously conjectured exact relation between spectral and connectivity dimensions on more general tree-like structures.