The spectral dimension of generic trees

TL;DR

This paper investigates the spectral dimension of certain infinite random trees, showing that for a broad class of these trees, the spectral dimension is 4/3, which is distinct from their Hausdorff dimension of 2.

Contribution

It introduces a new class of generic infinite trees and determines their spectral dimension and critical exponents, expanding understanding of their geometric and spectral properties.

Findings

Spectral dimension of these trees is 4/3.

Hausdorff dimension of the trees is 2.

Critical exponent of the mass is 1/3.

Abstract

We define generic ensembles of infinite trees. These are limits as $N\to\infty$ of ensembles of finite trees of fixed size $N$, defined in terms of a set of branching weights. Among these ensembles are those supported on trees with vertices of a uniformly bounded order. The associated probability measures are supported on trees with a single spine and Hausdorff dimension $d_h =2$. Our main result is that their spectral dimension is $d_s=4/3$, and that the critical exponent of the mass, defined as the exponential decay rate of the two-point function along the spine, is 1/3.