Friday, October 19, 2012

1002.2103 (Werner Kirsch et al.)

Lifshitz tails for a percolation model in the continuum    [PDF]

Werner Kirsch, Hatem Najar
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.
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