Entropy of random coverings and 4D quantum gravity

TL;DR

This paper analyzes the entropy of minimal geodesic ball coverings in manifolds and explores their implications for the continuum limit in discrete quantum gravity models, highlighting the role of fundamental group representations.

Contribution

It establishes conditions for exponential growth of coverings and links entropy estimates to Reidemeister torsion in quantum gravity contexts.

Findings

Entropy grows exponentially with volume under certain conditions.

Explicit entropy functions and bounds on critical exponents are derived.

Sum over inequivalent representations analyzed in 2D and 4D cases.

Abstract

We discuss the counting of minimal geodesic ball coverings of $n$-dimensional riemannian manifolds of bounded geometry, fixed Euler characteristic and Reidemeister torsion in a given representation of the fundamental group. This counting bears relevance to the analysis of the continuum limit of discrete models of quantum gravity. We establish the conditions under which the number of coverings grows exponentially with the volume, thus allowing for the search of a continuum limit of the corresponding discretized models. The resulting entropy estimates depend on representations of the fundamental group of the manifold through the corresponding Reidemeister torsion. We discuss the sum over inequivalent representations both in the two-dimensional and in the four-dimensional case. Explicit entropy functions as well as significant bounds on the associated critical exponents are obtained in both cases.