A mathematical model of the discrete 3-disk for the 3-dimensional Universe

A mathematical model of the distribution function for the discrete 3-disk is proposed in order to utilize in the statistical evolution equation of the 3-dimensional Universe. The model distribution is constructed based on analyses in known exact solutions of recursion equations for the generating functions of the discrete 2-disk.The proposed distribution function is compared with numerical simulations of the dynamical triangulation with $S^3$, and $D^3$ topologies.The model distribution exhibits three types of phases characterized by geometrical nature of the disk with either 1, 2, or 3- dimensional structure.Transitions between those phases are either cross-over, 1st order, or 2nd order depending on the parameters, which reflect the type of discretization and matter fields coupled to space.

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