  We investigate the spectral statistics of Hermitian matrices in which the
elements are chosen uniformly from U (1), called the uni-modular ensemble
(UME), in the limit of large matrix size. Using three complimentary methods; a
supersymmetric integration method, a combinatorial graph-theoretical analysis
and a Brownian motion approach, we are able to derive expressions for 1/N
corrections to the mean spectral moments and also analyse the fluctuations
about this mean. By addressing the same ensemble from three different point of
view, we can critically compare their relative advantages and derive some new
results.
