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In order to attain a statistical description of the evolution of cosmicdensity fluctuations in agreement with results from the numerical simulations,we introduce a probability conditional formalism (CF) based on an inventory ofisolated overdense regions in a density random field. This formalism is auseful tool for describing at the same time the mass function (MF) of darkhaloes, their mass aggregation histories (MAHs) and merging rates (MRs). The CFfocuses on virialized regions in a self-consistent way rather than in masselements, and it offers an economical description for a variety of randomfields. Within the framework of the CF, we confirm that, for a Gaussian field,it is not possible to reproduce at the same time the MF, MAH, and MR of haloes,both for a constant and moving barrier. Then, we develop an inductive methodfor constraining the cumulative conditional probability from a given halo MFdescription, and thus, using the CF, we calculate the halo MAHs and MRs. Byapplying this method to the MF measured in numerical simulations by Tinker etal. 2008, we find that a reasonable solution, justified by a mass conservationargument, is obtained if ones introduce a rescaling -increment by ~30% - of thevirial mass used in simulations and a (slight) deviation from Gaussianity.Thus, both the MAH and MR obtained by a Monte Carlo merger tree agree now withthe predictions of numerical simulations. We discuss on the necessity ofrescaling the virial mass in simulations when comparing with analyticalapproaches on the ground of the matter not accounted as part of the halos andthe halo mass limit due to numerical. Our analysis supports the presence of adiffuse dark matter component that is not taken into account in the measuredhalo MFs inasmuch as it is not part of the collapsed structures.

Mass function and assembly of dark halos: an approach to inventory isolated overdense regions in random fields