Peter Mueller


Gottingen (visiting UCI)


Thursday, October 21, 2004 - 2:00pm


MSTB 254

Bond-percolation graphs are random subgraphs of the d-dimensional
integer lattice generated by a standard Bernoulli bond-percolation
process. The
associated graph Laplacians, subject to Dirichlet or Neumann conditions at
cluster boundaries, represent bounded, self-adjoint, ergodic random
operators. They possess almost surely the
non-random spectrum [0,4d] and a self-averaging integrated density
of states. This integrated density of states is shown to exhibit Lifshits
tails at both spectral edges in the non-percolating phase. Depending
on the boundary condition and on the spectral edge, the Lifshits tail
discriminates between different cluster geometries (linear clusters
versus cube-like
clusters) which contribute the dominating eigenvalues. Lifshits tails
from cube-like clusters continue to show up above the percolation
In contrast, the other type of Lifshits tails cannot be observed in the
phase any more because they are hidden by van Hove singularities from the
percolating cluster.