A Random Necklace Model

TL;DR

This paper studies a Laplace operator on a random graph with infinitely many loops, showing that the integrated density of states exhibits discontinuities when the loop lengths have a nontrivial pure point distribution, supported by numerical examples.

Contribution

It introduces a novel random graph model with random loop lengths and analyzes the spectral properties of the associated Laplace operator.

Findings

Discontinuities in the integrated density of states occur with pure point distributions of loop lengths.

Numerical illustrations support the theoretical results.

Abstract

We consider a Laplace operator on a random graph consisting of infinitely many loops joined symmetrically by intervals of unit length. The arc lengths of the loops are considered to be independent, identically distributed random variables. The integrated density of states of this Laplace operator is shown to have discontinuities provided that the distribution of arc lengths of the loops has a nontrivial pure point part. Some numerical illustrations are also presented.