Title: Composition Dependent Instabilities in Mixtures With Many Components

Abstract: Understanding the phase behavior of mixtures with many components is important in many contexts, including as a key step toward a physics-based description of intracellular compartmentalization. Here, we study the instabilities of a mixture model where the second virial coefficients are taken as random Gaussian variables. Using tools from free probability theory we obtain the exact spinodal curve and the nature of instabilities for a mixture with an arbitrary composition, thus lifting the assumption of uniform mixture component densities pervading previous studies. We show that, by controlling the volume fraction of only a few components, one can systematically change the nature of the spinodal instability and achieve demixing for realistic scenarios by a strong {\em composition imbalance amplification}. This results from a non-trivial interplay of entropic effects due to non-uniform composition and complexity in the interactions. Our approach allows for the inclusion of any finite number of structured interactions, leading to a competition between different forms of demixing as density is varied.