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$(D+1)$-Colored Graphs – a Review of Sundry Properties
Symmetry, integrability and geometry: methods and applications, Tome 12 (2016) Cet article a éte moissonné depuis la source Math-Net.Ru

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We review the combinatorial, topological, algebraic and metric properties supported by $(D+1)$-colored graphs, with a focus on those that are pertinent to the study of tensor model theories. We show how to extract a limiting continuum metric space from this set of graphs and detail properties of this limit through the calculation of exponents at criticality.
Keywords: colored graph theory; random tensors; quantum gravity. 
@article{SIGMA_2016_12_a75,
     author = {James P. Ryan},
     title = {$(D+1)${-Colored} {Graphs} {\textendash} a {Review} of {Sundry} {Properties}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2016},
     volume = {12},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a75/}
}
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James P. Ryan. $(D+1)$-Colored Graphs – a Review of Sundry Properties. Symmetry, integrability and geometry: methods and applications, Tome 12 (2016). http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a75/

[1] Albenque M., Marckert J.-F., “Some families of increasing planar maps”, Electron. J. Probab., 13:56 (2008), 1624–1671, arXiv: 0712.0593 | DOI | MR | Zbl

[2] Ambjørn J., Durhuus B., Fröhlich J., “Diseases of triangulated random surface models, and possible cures”, Nuclear Phys. B, 257 (1985), 433–449 | DOI | MR

[3] Ambjørn J., Görlich A., Jurkiewicz J., Loll R., “Nonperturbative quantum gravity”, Phys. Rep., 519 (2012), 127–210, arXiv: 1203.3591 | DOI | MR

[4] Baratin A., Oriti D., “Group field theory with noncommutative metric variables”, Phys. Rev. Lett., 105 (2010), 221302, 4 pp., arXiv: 1002.4723 | DOI | MR

[5] Baratin A., Oriti D., “Group field theory and simplicial gravity path integrals: a model for Holst–Plebanski gravity”, Phys. Rev. D, 85 (2012), 044003, 15 pp., arXiv: 1111.5842 | DOI

[6] Benedetti D., Gurau R., “Phase transition in dually weighted colored tensor models”, Nuclear Phys. B, 855 (2012), 420–437, arXiv: 1108.5389 | DOI | MR | Zbl

[7] Bonzom V., Erbin H., “Coupling of hard dimers to dynamical lattices via random tensors”, J. Stat. Mech. Theory Exp., 2012 (2012), P09009, 18 pp., arXiv: 1204.3798 | DOI | MR

[8] Bonzom V., Gurau R., Riello A., Rivasseau V., “Critical behavior of colored tensor models in the large $N$ limit”, Nuclear Phys. B, 853 (2011), 174–195, arXiv: 1105.3122 | DOI | MR | Zbl

[9] Bonzom V., Gurau R., Rivasseau V., “The Ising model on random lattices in arbitrary dimensions”, Phys. Lett. B, 711 (2012), 88–96, arXiv: 1108.6269 | DOI | MR

[10] Bonzom V., Gurau R., Ryan J. P., Tanasa A., “The double scaling limit of random tensor models”, J. High Energy Phys., 2014:9 (2014), 051, 49 pp., arXiv: 1404.7517 | DOI | MR | Zbl

[11] Di Francesco P., Ginsparg P., Zinn-Justin J., “$2$D gravity and random matrices”, Phys. Rep., 254 (1995), 133, arXiv: hep-th/9306153 | DOI | MR

[12] Ferri M., Gagliardi C., “Crystallisation moves”, Pacific J. Math., 100 (1982), 85–103 | DOI | MR | Zbl

[13] Ferri M., Gagliardi C., Grasselli L., “A graph-theoretical representation of PL-manifolds — a survey on crystallizations”, Aequationes Math., 31 (1986), 121–141 | DOI | MR | Zbl

[14] Gurau R., “Lost in translation: topological singularities in group field theory”, Classical Quantum Gravity, 27 (2010), 235023, 20 pp., arXiv: 1006.0714 | DOI | MR | Zbl

[15] Gurau R., “Topological graph polynomials in colored group field theory”, Ann. Henri Poincaré, 11 (2010), 565–584, arXiv: 0911.1945 | DOI | MR | Zbl

[16] Gurau R., “Colored group field theory”, Comm. Math. Phys., 304 (2011), 69–93, arXiv: 0907.2582 | DOI | MR | Zbl

[17] Gurau R., “A generalization of the Virasoro algebra to arbitrary dimensions”, Nuclear Phys. B, 852 (2011), 592–614, arXiv: 1105.6072 | DOI | MR | Zbl

[18] Gurau R., “The $1/N$ expansion of colored tensor models”, Ann. Henri Poincaré, 12 (2011), 829–847, arXiv: 1011.2726 | DOI | MR | Zbl

[19] Gurau R., “The complete $1/N$ expansion of colored tensor models in arbitrary dimension”, Ann. Henri Poincaré, 13 (2012), 399–423, arXiv: 1102.5759 | DOI | MR | Zbl

[20] Gurau R., “The Schwinger–Dyson equations and the algebra of constraints of random tensor models at all orders”, Nuclear Phys. B, 865 (2012), 133–147, arXiv: 1203.4965 | DOI | MR | Zbl

[21] Gurau R., Rivasseau V., “The $1/N$ expansion of colored tensor models in arbitrary dimension”, Europhys. Lett., 95 (2011), 50004, 5 pp., arXiv: 1101.4182 | DOI | MR

[22] Gurau R., Ryan J. P., “Colored tensor models — a review”, SIGMA, 8 (2012), 020, 78 pp., arXiv: 1109.4812 | DOI | MR | Zbl

[23] Gurau R., Ryan J. P., “Melons are branched polymers”, Ann. Henri Poincaré, 15 (2014), 2085–2131, arXiv: 1302.4386 | DOI | MR | Zbl

[24] Gurau R., Schaeffer G., Regular colored graphs of positive degree, arXiv: 1307.5279

[25] Jonsson T., Wheater J. F., “The spectral dimension of the branched polymer phase of two-dimensional quantum gravity”, Nuclear Phys. B, 515 (1998), 549–574, arXiv: hep-lat/9710024 | DOI | MR | Zbl

[26] Le Gall J.-F., “Uniqueness and universality of the Brownian map”, Ann. Probab., 41 (2013), 2880–2960, arXiv: 1105.4842 | DOI | MR | Zbl

[27] Lins S., Gems, computers and attractors for $3$-manifolds, Series on Knots and Everything, 5, World Scientific Publishing Co., Inc., River Edge, NJ, 1995 | DOI | MR | Zbl

[28] Oriti D., “The quantum geometry of tensorial group field theories”, Symmetries and Groups in Contemporary Physics, Nankai Ser. Pure Appl. Math. Theoret. Phys., 11, World Sci. Publ., Hackensack, NJ, 2013, 379–384, arXiv: 1211.5714 | DOI | MR | Zbl

[29] Ryan J. P., “Tensor models and embedded Riemann surfaces”, Phys. Rev. D, 85 (2012), 024010, 9 pp., arXiv: 1104.5471 | DOI

[30] Tanasa A., “Multi-orientable group field theory”, J. Phys. A: Math. Theor., 45 (2012), 165401, 19 pp., arXiv: 1109.0694 | DOI | MR | Zbl

[31] Tanasa A., “The multi-orientable random tensor model, a review”, SIGMA, 12 (2016), 056, 23 pp., arXiv: 1512.02087 | DOI | MR | Zbl