Knill O.
Discrete random electromagnetic Laplacians
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ABSTRACT. We consider discrete random magnetic Laplacians in the plane and
discrete random electromagnetic Laplacians in higher dimensions.
The existence of these objects relies on
a theorem of Feldman-Moore which was
generalized by Lind to the nonabelian
case. For example, it allows to realize
ergodic Schr\"odinger operators
with stationary independent
magnetic fields on discrete two dimensional lattices including also
nonperiodic situations like Penrose lattices.
The theorem is generalized here to higher dimensions.
The Laplacians obtained from the electromagnetic vector
potential are elements of a
von Neumann algebra constructed from the underlying dynamical system
respectively from the ergodic equivalence relation. They generalize
Harper operators which correspond to constant magnetic fields.
For independent identically distributed
magnetic fields and special Anderson models,
we compute the density of states using a random walk expansion.