Expoential bounds on the number of causal triangulations

TL;DR

This paper establishes exponential bounds on the number of distinct causal triangulations of 3- and 4-spheres, advancing understanding of their combinatorial complexity in quantum gravity models.

Contribution

It proves exponential bounds on the count of causal triangulations in 3D and 4D, linking their growth to the bounds on triangulations of lower dimensions.

Findings

Number of 3D causal triangulations is exponentially bounded.

Number of 4D causal triangulations is exponentially bounded under certain conditions.

Provides a link between bounds in 3D and 4D triangulations.

Abstract

We prove that the number of combinatorially distinct causal 3-dimensional triangulations homeomorphic to the 3-dimensional sphere is bounded by an exponential function of the number of tetrahedra. It is also proven that the number of combinatorially distinct causal 4-dimensional triangulations homeomorphic to the 4-sphere is bounded by an exponential function of the number of 4-simplices provided the number of all combinatorially distinct triangulations of the 3-sphere is bounded by an exponential function of the number of tetrahedra.