Abstract
Melonic graphs constitute the family of graphs arising at leading order in the 1/N expansion of tensor models. They were shown to lead to a continuum phase, reminiscent of branched polymers. We show here that they are in fact precisely branched polymers, that is, they possess Hausdorff dimension 2 and spectral dimension 4/3.
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References
Gurau, R., Ryan, J.P.: Colored tensor models—a review. SIGMA 8, 020 (2012), (arXiv:1109.4812 [hep-th])
Di Francesco P., Ginsparg P.H., Zinn-Justin J.: 2-D gravity and random matrices. Phys. Rep. 254, 1–133 (1995)
Sasakura N.: Tensor model for gravity and orientability of manifold. Mod. Phys. Lett. A 6, 2613 (1991)
Ambjorn J., Durhuus B., Jonsson T.: Three-dimensional simplicial quantum gravity and generalized matrix models. Mod. Phys. Lett. A 6, 1133 (1991)
Sasakura, N.: Tensor models and 3-ary algebras. J. Math. Phys. 52, 103510 (2011), (arXiv:1104.1463 [hep-th])
Sasakura, N.: Tensor models and hierarchy of n-ary algebras. Int. J. Mod. Phys. A 26, 3249 (2011), (arXiv:1104.5312 [hep-th])
Boulatov, D.V.: A model of three-dimensional lattice gravity. Mod. Phys. Lett. A 7, 1629 (1992), (hep-th/9202074)
Ooguri, H.: Topological lattice models in four-dimensions. Mod. Phys. Lett. A 7, 2799 (1992), (hep-th/9205090)
De Pietri, R., Petronio, C.: Feynman diagrams of generalized matrix models and the associated manifolds in dimension 4. J. Math. Phys. 41, 6671 (2000), (gr-qc/0004045)
De Pietri, R., Freidel, L., Krasnov, K., Rovelli, C.: Barrett–Crane model from a Boulatov–Ooguri field theory over a homogeneous space. Nucl. Phys. B 574, 785 (2000), (hep-th/9907154)
Reisenberger, M.P., Rovelli, C.: Space-time as a Feynman diagram: the connection formulation. Class. Quant. Gravity 18, 121 (2001), (gr-qc/0002095)
Baratin, A., Oriti, D.: Group field theory with non-commutative metric variables. Phys. Rev. Lett. 105, 221302 (2010), (arXiv:1002.4723 [hep-th])
Baratin, A., Oriti, D.: Group field theory and simplicial gravity path integrals: a model for Holst–Plebanski gravity. Phys. Rev. D 85, 044003 (2012), (arXiv:1111.5842 [hep-th])
Rovelli C.: Quantum Gravity. Cambridge University Press, Cambridge (2004)
Engle, J., Livine, E., Pereira, R., Rovelli,C.: LQG vertex with finite Immirzi parameter. Nucl. Phys. B 799, 136 (2008), (arXiv:0711.0146 [gr-qc])
Freidel, L., Krasnov, K.: A new spin foam model for 4D Gravity. Class. Quant. Gravity 25, 125018 (2008), (arXiv:0708.1595 [gr-qc])
Livine, E.R., Speziale, S.: Consistently solving the simplicity constraints for spinfoam quantum gravity. Europhys. Lett. 81, 50004 (2008), (arXiv:0708.1915 [gr-qc])
Perini, C., Rovelli, C., Speziale, S.: Self-energy and vertex radiative corrections in LQG. Phys. Lett. B 682, 78 (2009), (arXiv:0810.1714 [gr-qc])
Ben Geloun, J., Gurau, R., Rivasseau, V.: EPRL/FK group field theory. Europhys. Lett. 92, 60008 (2010), (arXiv:1008.0354 [hep-th])
Krajewski, T., Magnen, J., Rivasseau, V., Tanasa, A., Vitale, P.: Quantum corrections in the group field theory formulation of the EPRL/FK models. Phys. Rev. D 82, 124069 (2010), (arXiv:1007.3150 [gr-qc])
Riello, A.: Self-energy of the Lorentzian EPRL-FK spin foam model of quantum gravity. Phys. Rev. D 88, 024011 (2013), (arXiv:1302.1781 [gr-qc])
Gurau, R.: Colored group field theory. Commun. Math. Phys. 304, 69 (2011), (arXiv:0907.2582 [hep-th])
Gurau, R.: The 1/N expansion of colored tensor models. Annales Henri Poincaré 12, 829 (2011), (arXiv:1011.2726 [gr-qc])
Gurau, R., Rivasseau, V.: The 1/N expansion of colored tensor models in arbitrary dimension. Europhys. Lett. 95, 50004 (2011), (arXiv:1101.4182 [gr-qc])
Gurau, R.: The complete 1/N expansion of colored tensor models in arbitrary dimension. Annales Henri Poincaré 13, 399 (2012), (arXiv:1102.5759 [gr-qc])
Bonzom, V.: New 1/N expansions in random tensor models. JHEP 1306, 062 (2013), (arXiv:1211.1657 [hep-th])
Dartois, S., Rivasseau, V., Tanasa, A.: The 1/N expansion of multi-orientable random tensor models. (arXiv:1301.1535 [hep-th])
Bonzom, V., Gurau, R., Riello, A., Rivasseau, V.: Critical behavior of colored tensor models in the large N limit. Nucl. Phys. B 853, 174 (2011), (arXiv:1105.3122 [hep-th])
Bonzom, V., Gurau, R., Rivasseau, V.: The Ising model on random lattices in arbitrary dimensions. Phys. Lett. B 711, 88 (2012), (arXiv:1108.6269 [hep-th])
Bonzom, V.: Multicritical tensor models and hard dimers on spherical random lattices. Phys. Lett. A 377, 501 (2013), (arXiv:1201.1931 [hep-th])
Bonzom, V., Erbin, H.: Coupling of hard dimers to dynamical lattices via random tensors. J. Stat. Mech. 1209, P09009 (2012), (arXiv:1204.3798 [cond-mat.stat-mech])
Benedetti, D., Gurau, R.: Phase transition in dually weighted colored tensor models. Nucl. Phys. B 855, 420 (2012), (arXiv:1108.5389 [hep-th])
Gurau, R.: A generalization of the Virasoro algebra to arbitrary dimensions. Nucl. Phys. B 852, 592 (2011), (arXiv:1105.6072 [hep-th])
Gurau, R.: The Schwinger Dyson equations and the algebra of constraints of random tensor models at all orders. Nucl. Phys. B 865, 133 (2012), (arXiv:1203.4965 [hep-th])
Krajewski, T.: Schwinger-Dyson equations in group field theories of quantum gravity. In: Proceedings of the XXIX international colloquium on group-theoretical methods in physics (GROUP 29), (arXiv:1211.1244 [math-ph])
Bonzom, V.: Revisiting random tensor models at large N via the Schwinger-Dyson equations. JHEP 1303, 160 (2013), (arXiv:1208.6216 [hep-th])
Gurau, R.: Universality for random tensors. To appear in Annales de l’Institut Henri Poincaré (B) Probabilités et Statistiques, (arXiv:1111.0519 [math.PR])
Ben Geloun, J., Rivasseau, V.: A renormalizable 4-dimensional tensor field theory. Commun. Math. Phys. 318, 69 (2013), (arXiv:1111.4997 [hep-th])
Ben Geloun, J.: Two and four-loop β-functions of rank 4 renormalizable tensor field theories. Class. Quant. Gravity 29, 235011 (2012), (arXiv:1205.5513 [hep-th])
Ben Geloun, J.: Asymptotic freedom of rank 4 tensor group field theory. (arXiv:1210.5490 [hep-th])
Carrozza, S., Oriti, D.: Bubbles and jackets: new scaling bounds in topological group field theories. JHEP 1206, 092 (2012), (arXiv:1203.5082 [hep-th])
Carrozza, S., Oriti, D., Rivasseau, V.: Renormalization of tensorial group field theories: abelian U(1) models in four dimensions. (arXiv:1207.6734 [hep-th])
Ambjorn J., Durhuus B., Jonsson T.: Summing over all genera for d > 1: a toy model. Phys. Lett. B 244, 403–412 (1990)
Bialas, P., Burda, Z.: Phase transition in fluctuating branched geometry. Phys. Lett. B 384, 75 (1996), (arXiv:hep-lat/9605020)
Ambjorn J., Durhuus B., Frohlich J.: Diseases of triangulated random surface models, and possible cures. Nucl. Phys. B 257, 433 (1985)
Ambjorn, J., Durhuus, B., Jonsson, T.: Quantum Geometry. A Statistical Field Theory Approach. University Press, Cambridge (Cambridge Monographs in Mathematical Physics), p. 363 (1997)
Ambjorn, J., Jurkiewicz, J., Loll, R.: Dynamically triangulating Lorentzian quantum gravity. Nucl. Phys. B 610, 347 (2001), (hep-th/0105267)
Jonsson, T., Wheater, J.F.: The spectral dimension of the branched polymer phase of two-dimensional quantum gravity. Nucl. Phys. B 515, 549 (1998), (hep-lat/9710024)
Albenque M., Marckert J.F.: Some families of increasing planar maps. Electron. J. Probab. 13(56), 1624–1671 (2008)
Pezzana M.: Sulla struttura topologica delle varietà compatte. Atti Sem. Mat. Fis. Univ. Modena 23, 269–277 (1974)
Ferri, M., Gagliardi, C.: Crystallisation moves. Pac. J. Math. 100, 1 (1982)
Gurau, R.: Lost in translation: topological singularities in group field theory. Class. Quant. Gravity 27, 235023 (2010), (arXiv:1006.0714 [hep-th])
Ryan, J.P.: Tensor models and embedded Riemann surfaces. Phys. Rev. D 85, 024010 (2012), (arXiv:1104.5471 [gr-qc])
Aldous D.: The continuum random tree II: an overview (Proceeding of the Durham symposium on stochastic analysis, 1990). Lond. Math. Soc. Lect. Note Ser. 167, 23–70 (1991)
Marckert J.F.: The lineage process in Galton–Watson trees and globally centered discrete snakes. Ann. Appl. Probab. 18(1), 209–244 (2009)
Durhuus, B., Jonsson, T., Wheater, J.F.: The spectral dimension of generic trees. (math-ph/0607020)
Bonzom, V., Gurau, R., Rivasseau, V.: Random tensor models in the large N limit: uncoloring the colored tensor models. Phys. Rev. D 85, 084037 (2012), (arXiv:1202.3637 [hep-th])
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Communicated by Abdelmalek Abdesselam.
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Gurau, R., Ryan, J.P. Melons are Branched Polymers. Ann. Henri Poincaré 15, 2085–2131 (2014). https://doi.org/10.1007/s00023-013-0291-3
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DOI: https://doi.org/10.1007/s00023-013-0291-3