Absolutely continuous spectrum for multi-type Galton Watson trees

TL;DR

This paper proves that for multi-type Galton Watson trees near finite cone type trees, the Laplace operator's spectrum is almost surely purely absolutely continuous, extending known spectral properties to a broader class of random trees.

Contribution

It establishes the presence of an almost sure purely absolutely continuous spectrum for Laplace operators on certain random multi-type Galton Watson trees, near finite cone type trees.

Findings

Spectrum is almost surely purely absolutely continuous.

The spectrum includes the absolutely continuous spectrum of the finite cone type tree.

Results apply to trees with at least one forward neighbor per vertex.

Abstract

We consider multi-type Galton Watson trees that are close to a tree of finite cone type in distribution. Moreover, we impose that each vertex has at least one forward neighbor. Then, we show that the spectrum of the Laplace operator exhibits almost surely a purely absolutely continuous component which is included in the absolutely continuous spectrum of the tree of finite cone type.