  The symmetrical restricted Gibbs ensemble (RGE) is a version of the Gibbs
ensemble in which particles are exchanged between two boxes of fixed equal
volumes. It has recently come to prominence because -- when combined with
specialized algorithms -- it provides for the study of near-coexistence density
fluctuations in highly size-asymmetric binary mixtures. Hitherto, however, a
detailed framework for extracting accurate estimates of critical point and
coexistence curve parameters from RGE density fluctuations has been lacking.
Here we address this problem by exploiting an exact link between the RGE
density fluctuations and those of the grand canonical ensemble. In the
sub-critical region we propose and test a simple method for obtaining accurate
estimates of coexistence densities. In the critical region we identify an
observable that serves as a finite system size estimator for the critical point
parameters, and present a finite-size scaling theory that allows extrapolation
to the thermodynamic limit.
