Random Holographic "Large Worlds" with Emergent Dimensions

Abstract

I propose a random network model governed by a Gaussian weight corresponding to Ising link antiferromagnetism as a model for emergent quantum space-time. In this model, discrete space is fundamental, not a regularization, its spectral dimension $d_s$ is not a model input but is, rather, completely determined by the antiferromagnetic coupling constant. Perturbative terms suppressing triangles and favouring squares lead to locally Euclidean ground states that are Ricci flat "large worlds" with power-law extension. I then consider the quenched graphs of lowest energy for $d_s=2$ and $d_s=3$ and I show how quenching leads to the spontaneous emergence of embedding spaces of Hausdorff dimension $d_H=4$ and $d_H=5$, respectively. One of the additional, spontaneous dimensions can be interpreted as time, causality being an emergent property that arises in the large $N$ limit (with $N$ the number of vertices). For $d_s=2$, the quenched graphs constitute a discrete version of a 5D-space-filling surface with a number of fundamental degrees of freedom scaling like $N^{2/5}$, a graph version of the holographic principle. These holographic degrees of freedom can be identified with the squares of the quenched graphs, which, being these triangle-free, are the fundamental area (or loop) quanta.