Localization on quantum graphs with random edge lengths

TL;DR

This paper investigates the spectral properties of quantum graphs with random edge lengths, demonstrating Anderson localization at the spectrum's bottom for a lattice structure with specific boundary conditions.

Contribution

It proves Anderson localization for quantum graphs with random edge lengths and non-zero vertex coupling, extending understanding of spectral behavior in random metric graphs.

Findings

Proves Anderson localization at the spectrum's bottom almost surely.

Shows localization occurs under non-vanishing vertex coupling.

Analyzes spectral edges beyond the bottom of the spectrum.

Abstract

The spectral properties of the Laplacian on a class of quantum graphs with random metric structure are studied. Namely, we consider quantum graphs spanned by the simple $\ZZ^d$-lattice with $\delta$-type boundary conditions at the vertices, and we assume that the edge lengths are randomly independently identically distributed. Under the assumption that the coupling constant at the vertices does not vanish, we show that the operator exhibits the Anderson localization at the bottom of the spectrum almost surely. We also study the case of other spectral edges.