On the Absence of an Exponential Bound in Four Dimensional Simplicial Gravity

TL;DR

This paper investigates a four-dimensional quantum gravity model based on triangulations, finding that the number of configurations grows faster than exponentially, implying the model lacks a thermodynamic limit.

Contribution

The study provides numerical evidence that the four-dimensional simplicial gravity model does not have an exponential bound on the number of triangulations, challenging its viability as a regularisation.

Findings

Number of triangulations grows faster than exponentially with volume

Model lacks a thermodynamic limit due to rapid growth

Numerical simulation confirms theoretical expectations

Abstract

We have studied a model which has been proposed as a regularisation for four dimensional quantum gravity. The partition function is constructed by performing a weighted sum over all triangulations of the four sphere. Using numerical simulation we find that the number of such triangulations containing $V$ simplices grows faster than exponentially with $V$. This property ensures that the model has no thermodynamic limit.