Lifshitz tails for a percolation model in the continuum

TL;DR

This paper investigates Lifshitz tails for the spectrum of the Laplacian in a continuum percolation model, revealing different spectral behaviors under Dirichlet and Neumann boundary conditions.

Contribution

It establishes Lifshitz tail behavior at the spectrum's bottom for Dirichlet conditions, independent of percolation probability, and analyzes bounds for Neumann conditions.

Findings

Lifshitz tails occur at the spectrum's bottom with Dirichlet boundary conditions.

Neumann boundary conditions lead to a lower bound characterized by van Hove behavior.

Lifshitz tails are proven regardless of percolation probability.

Abstract

In this paper we study Lifshitz tails for continuous Laplacian in a continuous site percolation situation. By this we mean that we delete a random set $\Gamma_\omega$ from $IR^d$ and consider the Dirichlet or Neumann Laplacian on $D=IR^d\setminus\Gamma_\omega$. We prove that the integrated density of states exhibits Lifshitz behavior at the bottom of the spectrum when we consider Dirichlet boundary conditions, while when we consider Neumann boundary conditions, it is bounded from below by a van Hove behavior. The Lifshitz tails are proven independently of the percolation probability, whereas for the van Hove case we need some assumption on the volume of the sets taken out as well as on the percolation probability.