  We propose a minimal model for miscibility phase transitions (MPTs) in a
class of asymmetric two-component heterogeneous fluid membranes at equilibrium
that generically display both first and second order MPTs, controlled by the
interplay of asymmetry and heterogeneity. In the vicinity of the MPTs, the
membrane fluctuations are generally enhanced. However, the degree of
enhancement is found to depend sensitively on the asymmetry-heterogeneity
coupling. We argue that experimental measurements of the membrane fluctuations
at the MPTs should provide physical information about the forms of the
asymmetry-heterogeneity couplings.
