Anderson Localization for Radial Tree Graphs With Random Branching Numbers

TL;DR

This paper proves Anderson localization for the discrete Laplace operator on radial tree graphs with random branching, using a novel approach involving direct sum representations and i.i.d. random variables.

Contribution

It introduces a new method to establish Anderson localization on radial trees with random branching numbers, expanding understanding of localization phenomena in complex graph structures.

Findings

Proves Anderson localization for the Laplace operator on radial trees with random branching.

Uses representation as direct sum of Jacobi matrices with i.i.d. singular distributions.

Establishes localization results for a class of graphs with random structure.

Abstract

We prove Anderson localization for the discrete Laplace operator on radial tree graphs with random branching numbers. Our method relies on the representation of the Laplace operator as the direct sum of half-line Jacobi matrices whose entries are non-degenerate, independent, identically distributed random variables with singular distributions.