The spectral dimension of random brushes

TL;DR

This paper investigates the spectral dimension of a class of random graphs called random brushes, showing that in two dimensions the spectral dimension is exactly 2, while in higher dimensions it varies within certain bounds.

Contribution

It provides rigorous bounds on the spectral dimension of random brushes across different dimensions, extending understanding of spectral properties of complex random graphs.

Findings

Spectral dimension in 2D is exactly 2.

In 3D, spectral dimension ranges between 2.5 and 3.

For dimensions 4 and higher, spectral dimension ranges between 3 and d.

Abstract

We consider a class of random graphs, called random brushes, which are constructed by adding linear graphs of random lengths to the vertices of Z^d viewed as a graph. We prove that for d=2 all random brushes have spectral dimension d_s=2. For d=3 we have {5\over 2}\leq d_s\leq 3 and for d\geq 4 we have 3\leq d_s\leq d.