Emergent time, cosmological constant and boundary dimension at infinity in combinatorial quantum gravity

TL;DR

This paper explores how combinatorial quantum gravity models lead to emergent space-time with a transition from negative to positive curvature, revealing a universe that avoids singularities and exhibits a boundary dimension of three at infinity.

Contribution

It provides a detailed picture of space-time emergence in combinatorial quantum gravity, linking discrete graph models to Lorentzian manifolds and cosmological constants.

Findings

Emergent space-time is modeled as a transition from negative to positive curvature.

The boundary dimension at infinity is always three for negative curvature manifolds.

The universe avoids a big bang singularity through a negative-curvature phase.

Abstract

Combinatorial quantum gravity is governed by a discrete Einstein-Hilbert action formulated on an ensemble of random graphs. There is strong evidence for a second-order quantum phase transition separating a random phase at strong coupling from an ordered, geometric phase at weak coupling. Here we derive the picture of space-time that emerges in the geometric phase, given such a continuous phase transition. In the geometric phase, ground-state graphs are discretizations of Riemannian, negative-curvature Cartan-Hadamard manifolds. On such manifolds, diffusion is ballistic. Asymptotically, diffusion time is soldered with a manifold coordinate and, consequently, the probability distribution is governed by the wave equation on the corresponding Lorentzian manifold of positive curvature, de Sitter space-time. With this asymptotic Lorentzian picture, the original negative curvature of the Riemannian manifold turns into a positive cosmological constant. The Lorentzian picture, however, is valid only asymptotically and cannot be extrapolated back in coordinate time. Before a certain epoch, coordinate time looses its meaning and the universe is a negative-curvature Riemannian "shuttlecock" with ballistic diffusion, thereby avoiding a big bang singularity. The so-called "dimension at infinity" of negative curvature manifolds, i.e. the large-scale spectral dimension seen by diffusion processes with no spectral gap, those that can probe the geometry at infinity, is always three.