Classical and quantum behavior of the integrated density of states for a randomly perturbed lattice

TL;DR

This paper analyzes the asymptotic behavior of the integrated density of states for a randomly perturbed lattice, revealing how classical and quantum effects dominate depending on the decay rate of the potential.

Contribution

It provides new estimates for the leading term of the integrated density of states considering different decay rates and multidimensional cases, extending known results from Poisson models.

Findings

Classical effect dominates with slow decay of potential.

Quantum effect appears with fast decay of potential.

Different asymptotic behaviors are identified in multidimensional cases.

Abstract

The asymptotic behavior of the integrated density of states for a randomly perturbed lattice at the infimum of the spectrum is investigated. The leading term is determined when the decay of the single site potential is slow. The leading term depends only on the classical effect from the scalar potential. To the contrary, the quantum effect appears when the decay of the single site potential is fast. The corresponding leading term is estimated and the leading order is determined. In the multidimensional cases, the leading order varies in different ways from the known results in the Poisson case. The same problem is considered for the negative potential. These estimates are applied to investigate the long time asymptotics of Wiener integrals associated with the random potentials.