Werner Kirsch, Peter Mueller
Spectral properties of the Laplacian on bond-percolation graphs
(206K, pdf)

ABSTRACT.  Bond-percolation graphs are random subgraphs of the d-dimensional 
integer lattice generated by a standard bond-percolation process. The 
associated graph Laplacians, subject to Dirichlet or Neumann conditions at 
cluster boundaries, represent bounded, self-adjoint, ergodic random 
operators with off-diagonal disorder. They possess almost surely the 
non-random spectrum [0,4d] and a self-averaging integrated density 
of states. The integrated density of states is shown to exhibit Lifshits 
tails at both spectral edges in the non-percolating phase. While the 
characteristic exponent of the Lifshits tail for the Dirichlet (Neumann) 
Laplacian at the lower (upper) spectral edge equals d/2, and thus depends 
on the spatial dimension, this is not the case at the upper (lower) spectral edge, where the exponent equals 1/2.