  We investigate the statistical properties of translation invariant random
fields (including point processes) on Euclidean spaces (or lattices) under
constraints on their spectrum or structure function. An important class of
models that motivate our study are hyperuniform and stealthy hyperuniform
systems, which are characterised by the vanishing of the structure function at
the origin (resp., vanishing in a neighbourhood of the origin). We show that
many key features of two classical statistical mechanical measures of
randomness - namely, fluctuations and entropy, are governed only by some
particular local aspects of their structure function. We obtain exponents for
the fluctuations of the local mass in domains of growing size, and show that
spatial geometric considerations play an important role - both the shape of the
domain and the mode of spectral decay. In doing so, we unveil intriguing
oscillatory behaviour of spatial correlations of local masses in adjacent box
domains. We describe very general conditions under which we show that the field
of local masses exhibit Gaussian asymptotics, with an explicitly described
limit. We further demonstrate that stealthy hyperuniform systems with joint
densities exhibit degeneracy in their asymptotic entropy per site. In fact, our
analysis shows that entropic degeneracy sets in under much milder conditions
than stealthiness, as soon as the structure function fails to be
logarithmically integrable.
