Convergence of Combinatorial Gravity

Kelly, Christy; Trugenberger, Carlo; Biancalana, Fabio

doi:10.48550/arxiv.2102.02356

Cited by 3 publications

(4 citation statements)

References 72 publications

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“…In the original proposal [6] the Ollivier combinatorial Ricci curvature [13][14][15] was used. This has been recently shown [16,17] to converge to standard continuum Ricci curvature on random geometric graphs [18] and is thus a genuine candidate for a discretization of general relativity. A simplified variant of Ollivier curvature has also been recently introduced in [19][20][21].…”

Section: Introductionmentioning

confidence: 86%

Enhanced Forman curvature and its relation to Ollivier curvature

Tee,

Trugenberger

2021

Preprint

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Recent advances in emergent geometry and discretized approaches to quantum gravity have relied upon the notion of a discrete measure of graph curvature. We focus on the two main measures that have been studied, the so-called Ollivier-Ricci and Forman-Ricci curvatures. These two approaches have a very different origin, and both have advantages and disadvantages. In this work we study the relationship between the two measures for a class of graphs that are important in quantum gravity applications. We discover that under a specific set of circumstances they are equivalent, potentially opening up the possibility of exploiting the relative strengths of both approaches in models of emergent spacetime and quantum gravity.

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Section: Introductionmentioning

confidence: 86%

Enhanced Forman curvature and its relation to Ollivier curvature

Tee,

Trugenberger

2021

Preprint

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“…In the original proposal [8] the Ollivier combinatorial Ricci curvature [16][17][18] was used. This has been recently shown [19,20] to converge to standard continuum Ricci curvature on random geometric graphs [21] and is thus a genuine candidate for a discretization of general relativity. A simplified variant of Ollivier curvature has also been recently introduced in [22][23][24].…”

mentioning

confidence: 86%

“…In the simplest realization the balls b i are chosen as unit balls, and the probability distributions μ i are uniform on these unit balls. Note, however, that the convergence to continuum Ricci curvature on generic random geometric graphs requires larger, "mesoscopic" balls, in order to "feel" the curvature of the background manifold [19,20]. Unit balls, instead, are sufficient in the flat case, when the curvature of the background manifold vanishes at all scales.…”

mentioning

confidence: 99%

Enhanced Forman curvature and its relation to Ollivier curvature

Tee¹,

Trugenberger²

2021

EPL

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Recent advances in emergent geometry and discretized approaches to quantum gravity have relied upon the notion of a discrete measure of graph curvature. We focus on the two main measures that have been studied, the so-called Ollivier-Ricci and Forman-Ricci curvatures. These two approaches have a very different origin, and both have advantages and disadvantages. In this work we study the relationship between the two measures for a class of graphs that are important in quantum gravity applications. We discover that under a specific set of circumstances they are equivalent, opening up the possibility of replacing the more fundamental Ollivier-Ricci curvature by the computationally more accessible Forman-Ricci curvature in certain applications to models of emergent spacetime and quantum gravity.

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“…Our philosophy here is different from that of "combinatorial quantum gravity"[33,34,35,36], which investigates the emergence of geometry from specific ensembles of random graphs, with the ambition of having its Lorentzian structure emerge alongside, something not yet achieved by any model as far as we know. In this approach, one uses the Ollivier-Ricci curvature in its original form for small values of δ, and has recently also compared it to another discrete notion of so-called Forman-Ricci curvature[37].…”

mentioning

confidence: 99%

Quantum Flatness in Two-Dimensional CDT Quantum Gravity

Brunekreef,

Loll

2021

Preprint

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Flatness -the absence of spacetime curvature -is a well-understood property of macroscopic, classical spacetimes in general relativity. The same cannot be said about the concepts of curvature and flatness in nonperturbative quantum gravity, where the microscopic structure of spacetime is not describable in terms of small fluctuations around a fixed background geometry. An interesting case are two-dimensional models of quantum gravity, which lack a classical limit and therefore are maximally "quantum". We investigate the recently introduced quantum Ricci curvature in CDT quantum gravity on a two-dimensional torus, whose quantum geometry could be expected to behave like a flat space on suitably coarsegrained scales. On the basis of Monte Carlo simulations we have performed, with system sizes of up to 600.000 building blocks, this does not seem to be the case. Instead, we find a scale-independent "quantum flatness", without an obvious classical analogue. As part of our study, we develop a criterion that allows us to distinguish between local and global, topological properties of the toroidal quantum system.

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Convergence of Combinatorial Gravity

Cited by 3 publications

References 72 publications

Enhanced Forman curvature and its relation to Ollivier curvature

Enhanced Forman curvature and its relation to Ollivier curvature

Enhanced Forman curvature and its relation to Ollivier curvature

Quantum Flatness in Two-Dimensional CDT Quantum Gravity

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