  $L$-ensembles are a class of determinantal point processes which can be
viewed as a statistical mechanical systems in the grand canonical ensemble.
Circulant $L$-ensembles are the subclass which are locally translationally
invariant and furthermore subject to periodic boundary conditions. Existing
theory can very simply be specialised to this setting, allowing for the
derivation of formulas for the system pressure, and the correlation kernel, in
the thermodynamic limit. For a one-dimensional domain, this is possible when
the circulant matrix is both real symmetric, or complex Hermitian. The special
case of the former having a Gaussian functional form for the entries is shown
to correspond to free fermions at finite temperature, and be generalisable to
higher dimensions. A special case of the latter is shown to be the statistical
mechanical model introduced by Gaudin to interpolate between Poisson and
unitary symmetry statistics in random matrix theory. It is shown in all cases
that the compressibility sum rule for the two-point correlation is obeyed, and
the small and large distance asymptotics of the latter are considered. Also, a
conjecture relating the asymptotic form of the hole probability to the pressure
is verified.
