Abstract
I propose a quantum gravity model in which geometric space emerges from random bits in a quantum phase transition driven by the combinatorial Ollivier-Ricci curvature and corresponding to the condensation of short cycles in random graphs. This quantum critical point defines quantum gravity non-perturbatively. In the ordered geometric phase at large distances the action reduces to the standard Einstein-Hilbert term.
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References
K. Becker, M. Becker and J. Schwarz, String Theory and M-Theory: A Modern Introduction, Cambridge University Press, Cambridge U.K. (2007).
M. Niedermaier and M. Reuters, The asymptotic safety scenario in quantum gravity, Living Rev. Relativ. 9 (2006) 5.
J. Ambjorn, A. Görlich, J. Jurkiewicz and R. Loll, Nonperturbative quantum gravity, Phys. Rept. 519 (2012) 127 [arXiv:1203.3591].
T. Regge, General relativity without coordinates, Nuovo Cim. 19 (1961) 558 [INSPIRE].
R.M. Williams and P.A. Tuckey, Regge calculus: a bibliography and brief review, Class. Quant. Grav. 9 (1992) 1409.
R. Albert and A.-L. Barabasi, Statistical mechanics of complex networks, Rev. Mod. Phys. 74 (2002) 47 [INSPIRE].
M. Penrose, Random geometric graphs, Oxford University Press, Oxford U.K. (2003).
J. Dall and M. Christensen, Random geometric graphs, Phys. Rev. E 66 (2002) 016121.
D. Krioukov, Clustering implies geometry in networks, Phys. Rev. Lett. 116 (2016) 208302.
Y. Ollivier, Ricci curvature of Markov chains on metric spaces, J. Funct. Anal. 256 (2009) 810.
Y. Ollivier, A survey of Ricci curvature for metric spaces and Markov chains, Adv. Stud. Pure Math. 57 (2010) 343.
Y. Lin, L. Lu and S.T. Yau, Ricci curvature of graphs, Tohoku Math. J. 63 (2011) 605.
B. Loisel and P. Romon, Ricci curvature on polyhedral surfaces via optimal transportation, Axioms 3 (2014) 119 [arXiv:1402.0644].
J. Jost and S. Liu, Ollivier’s Ricci curvature, local clustering and curvature inequalities on graphs, Discrete Comput. Geom. 51 (2014) 300.
B.B. Bhattacharya and S. Mukherjee, Exact and asymptotic results on coarse Ricci curvature of graphs, Discrete Math. 338 (2015) 23.
A. Perez, The spin-foam approach to quantum gravity, Living Rev. Relat. 16 (2013) 3.
D. Oriti, Group field theory and loop quantum gravity, in Loop quantum gravity: the first 30 years, J. Pullin et al. eds., World Scientific, Singapore (2014).
P.E. O’Neil, Asymptotics and random matrices with row-sum and column sum-restrictions, Bull. Am. Math. Soc. 75 (1969) 1276.
N.C. Wormald, Surveys in combinatorics, London Mathematical Society Lectures Note Series 267, Cambridge U.K. (1999).
F. Harary and B. Manvel, On the number of cycles in a graph, Mat. Casopis Slov. Akad. Vied 21 (1971) 55.
J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].
T. Jacobson, Thermodynamics of space-time: the Einstein equation of state, Phys. Rev. Lett. 75 (1995) 1260 [gr-qc/9504004] [INSPIRE].
T. Jacobson, Entanglement equilibrium and the Einstein equation, Phys. Rev. Lett. 116 (2016) 201101 [arXiv:1505.04753] [INSPIRE].
C.A. Trugenberger, Random holographic “large worlds” with emergent dimensions, Phys. Rev. E 94 (2016) 052305 [arXiv:1610.05339] [INSPIRE].
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Trugenberger, C.A. Combinatorial quantum gravity: geometry from random bits. J. High Energ. Phys. 2017, 45 (2017). https://doi.org/10.1007/JHEP09(2017)045
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DOI: https://doi.org/10.1007/JHEP09(2017)045