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

TL;DR

This paper introduces a mathematical model for the distribution of discrete 3-disks in 3D space, analyzing phase transitions and comparing with numerical simulations to understand the universe's geometric evolution.

Contribution

It develops a novel distribution function model for discrete 3-disks based on recursion equations, extending analysis from 2D to 3D and incorporating phase transition behaviors.

Findings

Model captures three geometric phases of the 3-disk.

Transitions include cross-over, first, and second order.

Model agrees with numerical simulations of dynamical triangulation.

Abstract

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.