Exact Hausdorff dimension of the spectral measure for the graph Laplacian on a sparse tree

TL;DR

This paper precisely determines the Hausdorff dimension of the spectral measure for the graph Laplacian on a sparse tree using an intermittency function, estimated via methods from one-dimensional discrete Schrödinger operators.

Contribution

It provides an exact formula for the Hausdorff dimension of the spectral measure on sparse trees, linking it to an intermittency function and employing Schrödinger operator techniques.

Findings

Exact Hausdorff dimension formula derived

Intermittency function estimated via Schrödinger methods

Spectral measure properties characterized precisely

Abstract

The Hausdorff dimension of spectral measure for the graph Laplacian is shown exactly in terms of an intermittency function. The intermittency function can be estimated by using one-dimensional discrete Schr\"{o}dinger operator method.