  We have studied the contact network properties of two and three dimensional
polydisperse, frictionless sphere packings at the random closed packing density
through simulations. We observe universal correlations between particle size
and contact number that are independent of the polydispersity of the packing.
This allows us to formulate a mean field version of the granocentric model to
predict the contact number distribution P(z). We find the predictions to be in
good agreement with a wide range of discrete and continuous size distributions.
The values of the two parameters that appear in the model are also independent
of the polydispersity of the packing. Finally we look at the nearest neighbour
spatial correlations to investigate the validity of the granocentric approach.
We find that both particle size and contact number are anti-correlated which
contrasts with the assumptions of the granocentric model. Despite this
shortcoming, the correlations are sufficiently weak which explains the good
approximation of P(z) obtained from the model.
