Asymptotic behavior of the density of states on a random lattice

TL;DR

This paper investigates the asymptotic eigenvalue distribution of the Laplacian on a random lattice with fluctuating connectivity, revealing localized states and a periodic structure in the density of states.

Contribution

It provides an analytical description of localized states and the asymptotic density of states, extending previous numerical findings with saddle point solutions that break rotational symmetry.

Findings

Localized states outside the mobility band are characterized by saddle point solutions.

The density of states exhibits a series of peaks with periodicity 1/q.

The model describes relaxation processes in media with geometrical defects.

Abstract

We study the diffusion of a particle on a random lattice with fluctuating local connectivity of average value q. This model is a basic description of relaxation processes in random media with geometrical defects. We analyze here the asymptotic behavior of the eigenvalue distribution for the Laplacian operator. We found that the localized states outside the mobility band and observed by Biroli and Monasson (1999, J. Phys. A: Math. Gen. 32 L255), in a previous numerical analysis, are described by saddle point solutions that breaks the rotational symmetry of the main action in the real space. The density of states is characterized asymptotically by a series of peaks with periodicity 1/q.