We extend our theory of amorphous packings of hard spheres to binary mixtures and more generally to multicomponent systems. The theory is based on the assumption that amorphous packings produced by typical experimental or numerical protocols can be identified with the infinite pressure limit of long lived metastable glassy states. We test this assumption against numerical and experimental data and show that the theory correctly reproduces the variation with mixture composition of structural observables, such as the total packing fraction and the partial coordination numbers.Amorphous packings of hard spheres are ubiquitous in physics: they have been used as models for liquids, glasses, colloidal systems, granular systems, and powders. They are also related to important problems in mathematics and information theory, such as digitalization of signals, error correcting codes, and optimization problems. Moreover, the structure and density (or porosity) of amorphous multicomponent packings is important in many branches of science and technology, ranging from oil extraction to storage of grains in silos.Despite being empirically studied since at least sixty years, amorphous packings still lack a precise mathematical definition, due to the intrinsic difficulty of quantifying "randomness" [1]. Indeed, even if a sphere packing is a purely geometrical object, in practice dense amorphous packings always result from rather complicated dynamical protocols: for instance, spheres can be thrown at random in a box that is subsequently shaken to achieve compactification [2], or they can be deposited onto a random seed cluster [3]. In numerical simulations, one starts from a random distribution of small spheres and inflates them until a jammed state is reached [4,5]; alternatively, one starts from large overlapping spheres and reduces the diameter in order to eliminate the overlaps [6][7][8][9]. In principle, each of these dynamical prescriptions produces an ensemble of final packings that depends on the details of the procedure used. Still, very remarkably, if the presence of crystalline regions is avoided, the structural properties of amorphous packings turn out to be very similar. This observation led to the proposal that "typical" amorphous packings should have common structure and density; the latter has been denoted Random Close Packing (RCP) density. The definition of RCP has been intensively debated in the last few years, in connection with the progresses of numerical simulations [1,10].Nevertheless, the empirical evidence, that amorphous packings produced according to very different protocols have common structural properties, is striking and call for an explanation. This is all the more true for binary or multicomponent mixtures, where in addition to the usual structural observables, such as the structure factor, one can investigate other quantities such as the coordination between spheres of different type, and study their variation with the composition of the mixture.In earlier attempts to build statistical models of ...

Abstract-Approximate message passing is an iterative algorithm for compressed sensing and related applications. A solid theory about the performance and convergence of the algorithm exists for measurement matrices having iid entries of zero mean. However, it was observed by several authors that for more general matrices the algorithm often encounters convergence problems. In this paper we identify the reason of the non-convergence for measurement matrices with iid entries and non-zero mean in the context of Bayes optimal inference. Finally we demonstrate numerically that when the iterative update is changed from parallel to sequential the convergence is restored.