### Abstract

We present a Euclidean quantum gravity model in which random graphs dynamically self-assemble into discrete manifold structures. Concretely, we consider a statistical model driven by a discretisation of the Euclidean Einstein-Hilbert action; contrary to previous approaches based on simplicial complexes and Regge calculus our discretisation is based on the Ollivier curvature, a coarse analogue of the manifold Ricci curvature defined for generic graphs. The Ollivier curvature is generally difficult to evaluate due to its definition in terms of optimal transport theory, but we present a new exact expression for the Ollivier curvature in a wide class of relevant graphs purely in terms of the numbers of short cycles at an edge. This result should be of independent intrinsic interest to network theorists. Action minimising configurations prove to be cubic complexes up to defects; there are indications that such defects are dynamically suppressed in the macroscopic limit. Closer examination of a defect free model shows that certain classical configurations have a geometric interpretation and discretely approximate vacuum solutions to the Euclidean Einstein-Hilbert action. Working in a configuration space where the geometric configurations are stable vacua of the theory, we obtain direct numerical evidence for the existence of a continuous phase transition; this makes the model a UV completion of Euclidean Einstein gravity. Notably, this phase transition implies an area-law for the entropy of emerging geometric space. Certain vacua of the theory can be interpreted as baby universes; we find that these configurations appear as stable vacua in a mean field approximation of our model, but are excluded dynamically whenever the action is exact indicating the dynamical stability of geometric space. The model is intended as a setting for subsequent studies of emergent time mechanisms.

Original language | English |
---|---|

Article number | 125012 |

Journal | Classical and Quantum Gravity |

Volume | 36 |

Issue number | 12 |

Early online date | 29 May 2019 |

DOIs | |

Publication status | Published - 20 Jun 2019 |

### Fingerprint

### Keywords

- discrete geometry
- emergent space
- Euclidean quantum gravity
- Ollivier curvature
- quantum gravity
- random graphs
- statistical mechanics

### ASJC Scopus subject areas

- Physics and Astronomy (miscellaneous)

### Cite this

*Classical and Quantum Gravity*,

*36*(12), [125012]. https://doi.org/10.1088/1361-6382/ab1c7d

}

*Classical and Quantum Gravity*, vol. 36, no. 12, 125012. https://doi.org/10.1088/1361-6382/ab1c7d

**Self-assembly of geometric space from random graphs.** / Kelly, Christy; Trugenberger, Carlo A.; Biancalana, Fabio.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Self-assembly of geometric space from random graphs

AU - Kelly, Christy

AU - Trugenberger, Carlo A.

AU - Biancalana, Fabio

PY - 2019/6/20

Y1 - 2019/6/20

N2 - We present a Euclidean quantum gravity model in which random graphs dynamically self-assemble into discrete manifold structures. Concretely, we consider a statistical model driven by a discretisation of the Euclidean Einstein-Hilbert action; contrary to previous approaches based on simplicial complexes and Regge calculus our discretisation is based on the Ollivier curvature, a coarse analogue of the manifold Ricci curvature defined for generic graphs. The Ollivier curvature is generally difficult to evaluate due to its definition in terms of optimal transport theory, but we present a new exact expression for the Ollivier curvature in a wide class of relevant graphs purely in terms of the numbers of short cycles at an edge. This result should be of independent intrinsic interest to network theorists. Action minimising configurations prove to be cubic complexes up to defects; there are indications that such defects are dynamically suppressed in the macroscopic limit. Closer examination of a defect free model shows that certain classical configurations have a geometric interpretation and discretely approximate vacuum solutions to the Euclidean Einstein-Hilbert action. Working in a configuration space where the geometric configurations are stable vacua of the theory, we obtain direct numerical evidence for the existence of a continuous phase transition; this makes the model a UV completion of Euclidean Einstein gravity. Notably, this phase transition implies an area-law for the entropy of emerging geometric space. Certain vacua of the theory can be interpreted as baby universes; we find that these configurations appear as stable vacua in a mean field approximation of our model, but are excluded dynamically whenever the action is exact indicating the dynamical stability of geometric space. The model is intended as a setting for subsequent studies of emergent time mechanisms.

AB - We present a Euclidean quantum gravity model in which random graphs dynamically self-assemble into discrete manifold structures. Concretely, we consider a statistical model driven by a discretisation of the Euclidean Einstein-Hilbert action; contrary to previous approaches based on simplicial complexes and Regge calculus our discretisation is based on the Ollivier curvature, a coarse analogue of the manifold Ricci curvature defined for generic graphs. The Ollivier curvature is generally difficult to evaluate due to its definition in terms of optimal transport theory, but we present a new exact expression for the Ollivier curvature in a wide class of relevant graphs purely in terms of the numbers of short cycles at an edge. This result should be of independent intrinsic interest to network theorists. Action minimising configurations prove to be cubic complexes up to defects; there are indications that such defects are dynamically suppressed in the macroscopic limit. Closer examination of a defect free model shows that certain classical configurations have a geometric interpretation and discretely approximate vacuum solutions to the Euclidean Einstein-Hilbert action. Working in a configuration space where the geometric configurations are stable vacua of the theory, we obtain direct numerical evidence for the existence of a continuous phase transition; this makes the model a UV completion of Euclidean Einstein gravity. Notably, this phase transition implies an area-law for the entropy of emerging geometric space. Certain vacua of the theory can be interpreted as baby universes; we find that these configurations appear as stable vacua in a mean field approximation of our model, but are excluded dynamically whenever the action is exact indicating the dynamical stability of geometric space. The model is intended as a setting for subsequent studies of emergent time mechanisms.

KW - discrete geometry

KW - emergent space

KW - Euclidean quantum gravity

KW - Ollivier curvature

KW - quantum gravity

KW - random graphs

KW - statistical mechanics

UR - http://www.scopus.com/inward/record.url?scp=85068638562&partnerID=8YFLogxK

U2 - 10.1088/1361-6382/ab1c7d

DO - 10.1088/1361-6382/ab1c7d

M3 - Article

AN - SCOPUS:85068638562

VL - 36

JO - Classical and Quantum Gravity

JF - Classical and Quantum Gravity

SN - 0264-9381

IS - 12

M1 - 125012

ER -