Invariant Correlations in Simplicial Gravity

TL;DR

This paper investigates invariant correlations in simplicial gravity with a combined cosmological and higher derivative action, revealing distinct behaviors for volume and curvature correlations and their decay patterns near the critical point.

Contribution

It presents the first numerical analysis of invariant correlations in simplicial gravity, highlighting different large-distance behaviors for volume and curvature correlations.

Findings

Volume correlations are negative definite at large distances.

Curvature correlations remain positive at large distances.

Correlations decay exponentially in the smooth phase and follow a power law near the critical point.

Abstract

Some first results are presented regarding the behavior of invariant correlations in simplicial gravity, with an action containing both a bare cosmological term and a lattice higher derivative term. The determination of invariant correlations as a function of geodesic distance by numerical methods is a difficult task, since the geodesic distance between any two points is a function of the fluctuating background geometry, and correlation effects become rather small for large distances. Still, a strikingly different behavior is found for the volume and curvature correlation functions. While the first one is found to be negative definite at large geodesic distances, the second one is always positive for large distances. For both correlations the results are consistent in the smooth phase with an exponential decay, turning into a power law close to the critical point at $G_c$. Such a behavior is not completely unexpected, if the model is to reproduce the classical Einstein theory at distances much larger than the ultraviolet cutoff scale.