The Cosmological Mass Distribution from Cayley Trees with Disorder

We present a new approach to the statistics of the cosmic density field and to the mass distribution of high-contrast structures, based on the formalism of Cayley trees. Our approach includes in one random process both fluctuations and interactions of the density perturbations. We connect tree-related quantities, like the partition function or its generating function, to the mass distribution. The Press \& Schechter mass function and the Smoluchowski kinetic equation are naturally recovered as two limiting cases corresponding to independent Gaussian fluctuations, and to aggregation of high-contrast condensations, respectively. Numerical realizations of the complete random process on the tree yield an excess of large-mass objects relative to the Press \& Schechter function. When interactions are fully effective, a power-law distribution with logarithmic slope -2 is generated.