Mixtures with a large number of components can undergo phase transitions of a hybrid character, with both condensation and demixing contributions. We describe a robust Monte Carlo simulation method for calculating phase coexistence in multicomponent mixtures. We use this approach to study the phase behavior of lattice models of multicomponent mixtures with strongly varying pair interactions. Such a system can be thought of as a simplified model of the cytosol, with both specific and nonspecific interactions. We show that mapping a multicomponent mixture onto an approximately equivalent one-component system yields both upper and lower bounds on the maximum solute volume fraction of a stable, homogeneous phase. By following the minimum excess-free-energy path from the dilute phase free-energy minimum, we predict the difference in composition between the condensed and dilute phases at the boundary of the homogeneous phase. We find that this "direction" of phase separation rarely aligns with the dominant direction of density fluctuations in the dilute phase. We also show that demixing transitions tend to lower the maximum solute volume fraction at which the homogeneous phase is stable. By considering statistical ensembles of mixtures with random interactions, we show that the demixing contribution to phase separation is self-averaging and dependent only on the mean and variance of the distribution of interactions.
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