  Mixtures with many components can segregate into coexisting phases, e.g., in
biological cells and synthetic materials such as metallic glass. The
interactions between components dictate what phases form in equilibrium, but
quantifying this relationship has proven difficult. We derive scaling relations
for the number of coexisting phases in multicomponent liquids with random
interactions and compositions, which we verify numerically. Our results
indicate that interactions only need to increase logarithmically with the
number of components for the liquid to segregate into many phases. In contrast,
a stability analysis of the homogeneous state predicts a power-law scaling.
This discrepancy implies an enormous parameter regime where the number of
coexisting phases exceeds the number of unstable modes, generalizing the
nucleation and growth regime of binary mixtures to many components.
