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We present a model, that while simple to define, shows rich behavior
that accounts for a variety of physical phenomena, such as
localization of phonons, anomalous diffusion, and slow relaxations in
glassy systems. The 'crux of the matter' lies in the fact that the
rate of many process es in nature is exponential in the relevant
distance: quantum tunneling is a common example. If we think about a
particle diffusing in a random environment, its dynamics will be
described by a matrix A where the i,j'th element is
exponential in the distance between points i and j. This leads us
naturally to the model of exponential random matrices, which is a
different ensemble of random matrices, with interesting properties. We
solve the model exactly in the low density, and discuss the
implications on the various physical problems.
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