Invitation to random tensors. (English) Zbl 1346.83030
Summary: This article is preface to the SIGMA special issue “Tensor Models, Formalism and Applications”, http://www.emis.de/journals/SIGMA/Tensor_Models.html. The issue is a collection of eight excellent, up to date reviews on random tensor models. The reviews combine pedagogical introductions meant for a general audience with presentations of the most recent developments in the field. This preface aims to give a condensed panoramic overview of random tensors as the natural generalization of random matrices to higher dimensions.
MSC:
| 83C45 | Quantization of the gravitational field |
Keywords:
random tensorsReferences:
| [1] | Ambj{\o}rn, Jan and Durhuus, Bergfinnur and J{\'o}nsson, Th{\'o}rdur, Three-dimensional simplicial quantum gravity and generalized matrix models, Modern Physics Letters A. Particles and Fields, Gravitation, Cosmology, Nuclear Physics, 6, 12, 1133-1146, (1991) · Zbl 1020.83537 · doi:10.1142/S0217732391001184 |
| [2] | Ambj{\o}rn, Jan and Durhuus, Bergfinnur and Jonsson, Thordur, Quantum geometry. A statistical field theory approach, Cambridge Monographs on Mathematical Physics, xiv+363, (1997), Cambridge University Press, Cambridge · Zbl 0993.82500 · doi:10.1017/CBO9780511524417 |
| [3] | Ambj{\o}rn, J. and Jurkiewicz, J. and Makeenko, Yu. M., Multiloop correlators for two-dimensional quantum gravity, Physics Letters. B, 251, 4, 517-524, (1990) · doi:10.1016/0370-2693(90)90790-D |
| [4] | Baratin, Aristide and Carrozza, Sylvain and Oriti, Daniele and Ryan, James and Smerlak, Matteo, Melonic phase transition in group field theory, Letters in Mathematical Physics. A Journal for the Rapid Dissemination of Short Contributions in the Field of Mathematical Physics, 104, 8, 1003-1017, (2014) · Zbl 1297.81136 · doi:10.1007/s11005-014-0699-9 |
| [5] | Baratin, Aristide and Oriti, Daniele, Group field theory with noncommutative metric variables, Physical Review Letters, 105, 22, 221302, 4 pages, (2010) · doi:10.1103/PhysRevLett.105.221302 |
| [6] | Baratin, Aristide and Oriti, Daniele, Ten questions on group field theory (and their tentative answers, Journal of Physics: Conference Series, 360, 012002, 10 pages, (2012) · doi:10.1088/1742-6596/360/1/012002 |
| [7] | Ben Arous, G. and Guionnet, A., Large deviations for {W}igner’s law and {V}oiculescu’s non-commutative entropy, Probability Theory and Related Fields, 108, 4, 517-542, (1997) · Zbl 0954.60029 · doi:10.1007/s004400050119 |
| [8] | Ben Geloun, J., Asymptotic freedom of rank 4 tensor group field theory, Symmetries and Groups in Contemporary Physics, Nankai Ser. Pure Appl. Math. Theoret. Phys., 11, 367-372, (2013), World Sci. Publ., Hackensack, NJ · Zbl 1298.83041 · doi:10.1142/9789814518550_0049 |
| [9] | Ben Geloun, Joseph, Two- and four-loop {\( \beta \)}-functions of rank-4 renormalizable tensor field theories, Classical and Quantum Gravity, 29, 23, 235011, 40 pages, (2012) · Zbl 1258.83033 · doi:10.1088/0264-9381/29/23/235011 |
| [10] | Ben Geloun, Joseph, On the finite amplitudes for open graphs in {A}belian dynamical colored {B}oulatov–{O}oguri models, Journal of Physics. A. Mathematical and Theoretical, 46, 40, 402002, 12 pages, (2013) · Zbl 1277.83040 · doi:10.1088/1751-8113/46/40/402002 |
| [11] | Ben Geloun, Joseph, Renormalizable models in rank {\(d\geq 2\)} tensorial group field theory, Communications in Mathematical Physics, 332, 1, 117-188, (2014) · Zbl 1300.83043 · doi:10.1007/s00220-014-2142-6 |
| [12] | Ben Geloun, Joseph and Koslowski, Tim A., Nontrivial {UV} behavior of rank-4 tensor field models for quantum gravity, (None) |
| [13] | Ben Geloun, Joseph and Livine, Etera R., Some classes of renormalizable tensor models, Journal of Mathematical Physics, 54, 8, 082303, 25 pages, (2013) · Zbl 1287.83021 · doi:10.1063/1.4818797 |
| [14] | Ben Geloun, Joseph and Magnen, Jacques and Rivasseau, Vincent, Bosonic colored group field theory, The European Physical Journal C. Particles and Fields, 70, 4, 1119-1130, (2010) · doi:10.1140/epjc/s10052-010-1487-z |
| [15] | Ben Geloun, Joseph and Martini, R. and Oriti, Daniele, Functional renormalization group analysis of tensorial group field theories on \(\mathbb{R}^d\), Physical Review D. Particles, Fields, Gravitation, and Cosmology, 94, 2, 024017, 45 pages, (2016) · doi:10.1103/PhysRevD.94.024017 |
| [16] | Ben Geloun, Joseph and Ramgoolam, Sanjaye, Counting tensor model observables and branched covers of the 2-sphere, Annales de l’Institut Henri Poincar\'e D. Combinatorics, Physics and their Interactions, 1, 1, 77-138, (2014) · Zbl 1288.15031 · doi:10.4171/AIHPD/4 |
| [17] | Ben Geloun, Joseph and Rivasseau, Vincent, A renormalizable 4-dimensional tensor field theory, Communications in Mathematical Physics, 318, 1, 69-109, (2013) · Zbl 1261.83016 · doi:10.1007/s00220-012-1549-1 |
| [18] | Ben Geloun, Joseph and Samary, Dine Ousmane, 3{D} tensor field theory: renormalization and one-loop {\( \beta \)}-functions, Annales Henri Poincar\'e. A Journal of Theoretical and Mathematical Physics, 14, 6, 1599-1642, (2013) · Zbl 1272.83028 · doi:10.1007/s00023-012-0225-5 |
| [19] | Ben Geloun, Joseph and Toriumi, Reiko, Parametric representation of rank {\(d\)} tensorial group field theory: {A}belian models with kinetic term {\( \sum_s\vert p_s\vert +\mu \)}, Journal of Mathematical Physics, 56, 9, 093503, 53 pages, (2015) · Zbl 1322.83011 · doi:10.1063/1.4929771 |
| [20] | Benedetti, Dario and Ben Geloun, Joseph and Oriti, Daniele, Functional renormalisation group approach for tensorial group field theory: a rank-3 model, Journal of High Energy Physics, 2015, 3, no. 3, 084, 40 pages, (2015) · Zbl 1388.83088 · doi:10.1007/JHEP03(2015)084 |
| [21] | Benedetti, Dario and Gurau, Razvan, Phase transition in dually weighted colored tensor models, Nuclear Physics. B, 855, 2, 420-437, (2012) · Zbl 1229.81208 · doi:10.1016/j.nuclphysb.2011.10.015 |
| [22] | Benedetti, Dario and Lahoche, Vincent, Functional renormalization group approach for tensorial group field theory: a rank-6 model with closure constraint, Classical and Quantum Gravity, 33, 9, 095003, 35 pages, (2016) · Zbl 1338.83069 · doi:10.1088/0264-9381/33/9/095003 |
| [23] | Bonzom, Valentin, Multi-critical tensor models and hard dimers on spherical random lattices, Physics Letters. A, 377, 7, 501-506, (2013) · Zbl 1428.82024 · doi:10.1016/j.physleta.2012.12.022 |
| [24] | Bonzom, Valentin, New {\(1/N\)} expansions in random tensor models, Journal of High Energy Physics, 2013, 6, no. 6, 062, 25 pages, (2013) · Zbl 1342.83053 · doi:10.1007/JHEP06(2013)062 |
| [25] | Bonzom, Valentin, Revisiting random tensor models at large {\(N\)} via the {S}chwinger–{D}yson equations, Journal of High Energy Physics, 2013, 3, no. 3, 160, 25 pages, (2013) · Zbl 1342.81174 · doi:10.1007/JHEP03(2013)160 |
| [26] | Bonzom, Valentin and Combes, Fr{\'e}d{\'e}ric, Tensor models from the viewpoint of matrix models: the cases of loop models on random surfaces and of the {G}aussian distribution, Annales de l’Institut Henri Poincar\'e D. Combinatorics, Physics and their Interactions, 2, 1, 1-47, (2015) · Zbl 1310.05084 · doi:10.4171/AIHPD/14 |
| [27] | Bonzom, Valentin and Erbin, Harold, Coupling of hard dimers to dynamical lattices via random tensors, Journal of Statistical Mechanics: Theory and Experiment, 2012, 9, P09009, 18 pages, (2012) · Zbl 1456.82095 · doi:10.1088/1742-5468/2012/09/P09009 |
| [28] | Bonzom, Valentin and Gurau, Razvan and Riello, Aldo and Rivasseau, Vincent, Critical behavior of colored tensor models in the large {\(N\)} limit, Nuclear Physics. B, 853, 1, 174-195, (2011) · Zbl 1229.81222 · doi:10.1016/j.nuclphysb.2011.07.022 |
| [29] | Bonzom, Valentin and Gurau, Razvan and Rivasseau, Vincent, Random tensor models in the large \(N\) limit: uncoloring the colored tensor models, Physical Review D. Particles, Fields, Gravitation, and Cosmology, 85, 8, 084037, 12 pages, (2012) · doi:10.1103/PhysRevD.85.084037 |
| [30] | Bonzom, Valentin and Gurau, Razvan and Rivasseau, Vincent, The {I}sing model on random lattices in arbitrary dimensions, Physics Letters. B, 711, 1, 88-96, (2012) · doi:10.1016/j.physletb.2012.03.054 |
| [31] | Bonzom, Valentin and Gurau, Razvan and Ryan, James P. and Tanasa, Adrian, The double scaling limit of random tensor models, Journal of High Energy Physics, 2014, 9, no. 9, 051, 49 pages, (2014) · Zbl 1333.60014 · doi:10.1007/JHEP09(2014)051 |
| [32] | Boulatov, D. V., A model of three-dimensional lattice gravity, Modern Physics Letters A. Particles and Fields, Gravitation, Cosmology, Nuclear Physics, 7, 18, 1629-1646, (1992) · Zbl 1020.83539 · doi:10.1142/S0217732392001324 |
| [33] | Boulatov, D. V. and Kazakov, V. A., The {I}sing model on a random planar lattice: the structure of the phase transition and the exact critical exponents, Physics Letters. B, 186, 3-4, 379-384, (1987) · doi:10.1016/0370-2693(87)90312-1 |
| [34] | Br{\'e}zin, {\'E}douard and Douglas, Michael R. and Kazakov, Vladimir and Shenker, Stephen H., The {I}sing model coupled to {\(2\)}{D} gravity. {A} nonperturbative analysis, Physics Letters. B, 237, 1, 43-46, (1990) · doi:10.1016/0370-2693(90)90458-I |
| [35] | Br{\'e}zin, E. and Itzykson, C. and Parisi, G. and Zuber, J. B., Planar diagrams, Communications in Mathematical Physics, 59, 1, 35-51, (1978) · Zbl 0997.81548 · doi:10.1007/BF01614153 |
| [36] | Br{\'e}zin, E. and Kazakov, V. A., Exactly solvable field theories of closed strings, Physics Letters. B, 236, 2, 144-150, (1990) · doi:10.1016/0370-2693(90)90818-Q |
| [37] | Carrozza, Sylvain, Discrete renormalization group for {\({\rm SU}(2)\)} tensorial group field theory, Annales de l’Institut Henri Poincar\'e D. Combinatorics, Physics and their Interactions, 2, 1, 49-112, (2015) · Zbl 1319.81068 · doi:10.4171/AIHPD/15 |
| [38] | Carrozza, Sylvain, Group field theory in dimension {\(4-\varepsilon \)}, Physical Review D. Particles, Fields, Gravitation, and Cosmology, 91, 6, 065023, 10 pages, (2015) · doi:10.1103/PhysRevD.91.065023 |
| [39] | Carrozza, Sylvain and Oriti, Daniele, Bounding bubbles: the vertex representation of \(3d\) group field theory and the suppression of pseudomanifolds, Physical Review D. Particles, Fields, Gravitation, and Cosmology, 85, 4, 044004, 22 pages, (2012) · doi:10.1103/PhysRevD.85.044004 |
| [40] | Carrozza, Sylvain and Oriti, Daniele, Bubbles and jackets: new scaling bounds in topological group field theories, Journal of High Energy Physics, 2012, 6, no. 6, 092, 42 pages, (2012) · Zbl 1397.81324 · doi:10.1007/JHEP06(2012)092 |
| [41] | Carrozza, Sylvain and Oriti, Daniele and Rivasseau, Vincent, Renormalization of a {\({\rm SU}(2)\)} tensorial group field theory in three dimensions, Communications in Mathematical Physics, 330, 2, 581-637, (2014) · Zbl 1300.83023 · doi:10.1007/s00220-014-1928-x |
| [42] | Carrozza, Sylvain and Oriti, Daniele and Rivasseau, Vincent, Renormalization of tensorial group field theories: {A}belian {\({\rm U}(1)\)} models in four dimensions, Communications in Mathematical Physics, 327, 2, 603-641, (2014) · Zbl 1291.83102 · doi:10.1007/s00220-014-1954-8 |
| [43] | Chapuy, Guillaume and Marcus, Michel and Schaeffer, Gilles, A bijection for rooted maps on orientable surfaces, SIAM Journal on Discrete Mathematics, 23, 3, 1587-1611, (2009) · Zbl 1207.05087 · doi:10.1137/080720097 |
| [44] | Cori, Robert and Schaeffer, Gilles, Description trees and {T}utte formulas, Theoretical Computer Science, 292, 1, 165-183, (2003) · Zbl 1063.68076 · doi:10.1016/S0304-3975(01)00221-3 |
| [45] | Dartois, St{\'e}phane and Gurau, Razvan and Rivasseau, Vincent, Double scaling in tensor models with a quartic interaction, Journal of High Energy Physics, 2013, 9, no. 9, 088, 33 pages, (2013) · Zbl 1342.83079 · doi:10.1007/JHEP09(2013)088 |
| [46] | Dartois, St{\'e}phane and Rivasseau, Vincent and Tanasa, Adrian, The {\(1/N\)} expansion of multi-orientable random tensor models, Annales Henri Poincar\'e. A Journal of Theoretical and Mathematical Physics, 15, 5, 965-984, (2014) · Zbl 1288.81070 · doi:10.1007/s00023-013-0262-8 |
| [47] | David, F., Planar diagrams, two-dimensional lattice gravity and surface models, Nuclear Physics. B, 257, 1, 45-58, (1985) · doi:10.1016/0550-3213(85)90335-9 |
| [48] | David, F., Conformal field theories coupled to {\(2\)}-{D} gravity in the conformal gauge, Modern Physics Letters A. Particles and Fields, Gravitation, Cosmology, Nuclear Physics, 3, 17, 1651-1656, (1988) · doi:10.1142/S0217732388001975 |
| [49] | David, F., Simplicial quantum gravity and random lattices, Gravitation and Quantizations ({L}es {H}ouches, 1992), 679-749, (1995), Amsterdam · Zbl 0856.53069 |
| [50] | Di Francesco, P. and Ginsparg, P. and Zinn-Justin, J., {\(2\)}{D} gravity and random matrices, Physics Reports. A Review Section of Physics Letters, 254, 1-2, 1-133, (1995) · doi:10.1016/0370-1573(94)00084-G |
| [51] | Dijkgraaf, Robbert and Verlinde, Herman and Verlinde, Erik, Loop equations and {V}irasoro constraints in nonperturbative two-dimensional quantum gravity, Nuclear Physics. B, 348, 3, 435-456, (1991) · doi:10.1016/0550-3213(91)90199-8 |
| [52] | Disertori, Margherita and Gurau, Razvan and Magnen, Jacques and Rivasseau, Vincent, Vanishing of beta function of non-commutative {\( \Phi_4^4\)} theory to all orders, Physics Letters. B, 649, 1, 95-102, (2007) · Zbl 1248.81253 · doi:10.1016/j.physletb.2007.04.007 |
| [53] | Distler, Jacques and Kawai, Hikaru, Conformal field theory and {\(2\)}{D} quantum gravity, Nuclear Physics. B, 321, 2, 509-527, (1989) · doi:10.1016/0550-3213(89)90354-4 |
| [54] | Douglas, Michael R. and Shenker, Stephen H., Strings in less than one dimension, Nuclear Physics. B, 335, 3, 635-654, (1990) · doi:10.1016/0550-3213(90)90522-F |
| [55] | Duplantier, Bertrand, Conformal random geometry, Mathematical Statistical Physics, 101-217, (2006), Elsevier B.V., Amsterdam · Zbl 1370.60013 · doi:10.1016/S0924-8099(06)80040-5 |
| [56] | Eynard, Bertrand, Topological expansion for the 1-{H}ermitian matrix model correlation functions, Journal of High Energy Physics. A SISSA Journal, 2004, 11, no. 11, 031, 35 pages, (2004) · doi:10.1088/1126-6708/2004/11/031 |
| [57] | Eynard, Bertrand, Another algebraic variational principle for the spectral curve of matrix models, (None) · Zbl 1226.82029 |
| [58] | Eynard, Bertrand and Orantin, N., Invariants of algebraic curves and topological expansion, Communications in Number Theory and Physics, 1, 2, 347-452, (2007) · Zbl 1161.14026 · doi:10.4310/CNTP.2007.v1.n2.a4 |
| [59] | Fukuma, Masafumi and Kawai, Hikaru and Nakayama, Ryuichi, Continuum {S}chwinger–{D}yson equations and universal structures in two-dimensional quantum gravity, International Journal of Modern Physics A. Particles and Fields. Gravitation. Cosmology. Nuclear Physics, 6, 8, 1385-1406, (1991) · doi:10.1142/S0217751X91000733 |
| [60] | Fusy, Eric and Tanasa, Adrian, Asymptotic expansion of the multi-orientable random tensor model, Electronic Journal of Combinatorics, 22, 1, 1.52, 30 pages, (2015) · Zbl 1310.81117 |
| [61] | Gielen, Steffen, Identifying cosmological perturbations in group field theory condensates, Journal of High Energy Physics, 2015, 8, no. 8, 010, 23 pages, (2015) · Zbl 1388.83916 · doi:10.1007/JHEP08(2015)010 |
| [62] | Gielen, Steffen, Perturbing a quantum gravity condensate, Physical Review D. Particles, Fields, Gravitation, and Cosmology, 91, 4, 043526, 11 pages, (2015) · doi:10.1103/PhysRevD.91.043526 |
| [63] | Gielen, Steffen, Emergence of a low spin phase in group field theory condensates, (None) · Zbl 1351.83021 |
| [64] | Gielen, Steffen and Oriti, Daniele, Quantum cosmology from quantum gravity condensates: cosmological variables and lattice-refined dynamics, New Journal of Physics, 16, December, 123004, 11 pages, (2014) · Zbl 1451.85010 · doi:10.1088/1367-2630/16/12/123004 |
| [65] | Gielen, Steffen and Oriti, Daniele and Sindoni, Lorenzo, Cosmology from group field theory formalism for quantum gravity, Physical Review Letters, 111, 3, 031301, 4 pages, (2013) · doi:10.1103/PhysRevLett.111.031301 |
| [66] | Gielen, Steffen and Oriti, Daniele and Sindoni, Lorenzo, Homogeneous cosmologies as group field theory condensates, Journal of High Energy Physics, 2014, 6, no. 6, 013, 69 pages, (2014) · Zbl 1333.81187 · doi:10.1007/JHEP06(2014)013 |
| [67] | Glashow, Sheldon L., Partial-symmetries of weak interactions, Nuclear Physics, 22, 4, 579-588, (1961) · doi:10.1016/0029-5582(61)90469-2 |
| [68] | Glimm, James and Jaffe, Arthur, Quantum physics. A functional integral point of view, xxii+535, (1987), Springer-Verlag, New York · doi:10.1007/978-1-4612-4728-9 |
| [69] | Goroff, Marc H. and Sagnotti, Augusto, The ultraviolet behavior of {E}instein gravity, Nuclear Physics. B, 266, 3-4, 709-736, (1986) · doi:10.1016/0550-3213(86)90193-8 |
| [70] | Gross, David J. and Migdal, Alexander A., Nonperturbative two-dimensional quantum gravity, Physical Review Letters, 64, 2, 127-130, (1990) · Zbl 1050.81610 · doi:10.1103/PhysRevLett.64.127 |
| [71] | Gross, David J. and Wilczek, Frank, Asymptotically free gauge theories. I, Physical Review. D, 8, 10, 3633-3652, (1973) · doi:10.1103/PhysRevD.8.3633 |
| [72] | Gross, David J. and Wilczek, Frank, Asymptotically free gauge theories. II, Physical Review. D, 9, 4, 980-993, (1974) · doi:10.1103/PhysRevD.9.980 |
| [73] | Gross, David J. and Wilczek, Frank, Ultraviolet behavior of nonabelian gauge theories, Physical Review Letters, 30, 26, 1343-1346, (1973) · doi:10.1103/PhysRevLett.30.1343 |
| [74] | Gross, Mark, Tensor models and simplicial quantum gravity in {\(>2\)}-{D}, Nuclear Physics B. Proceedings Supplement, 25A, 144-149, (1992) · Zbl 0957.83511 · doi:10.1016/S0920-5632(05)80015-5 |
| [75] | Grosse, Harald and Wulkenhaar, Raimar, Renormalisation of {\( \phi^4\)}-theory on noncommutative {\({\mathbb R}^4\)} in the matrix base, Communications in Mathematical Physics, 256, 2, 305-374, (2005) · Zbl 1075.82005 · doi:10.1007/s00220-004-1285-2 |
| [76] | Grosse, Harald and Wulkenhaar, Raimar, Progress in solving a noncommutative quantum field theory in four dimensions, (None) · Zbl 1305.81129 |
| [77] | Grosse, Harald and Wulkenhaar, Raimar, Construction of the {\( \Phi^4_4\)}-quantum field theory on noncommutative {M}oyal space, RIMS K\=oky\=uroku, 1904, 67-104, (2013) |
| [78] | Grosse, Harald and Wulkenhaar, Raimar, Solvable limits of a {\(4D\)} noncommutative {QFT}, (None) · Zbl 1334.81077 |
| [79] | Grosse, Harald and Wulkenhaar, Raimar, Solvable {4D} noncommutative {QFT}: phase transitions and quest for reflection positivity, (None) · Zbl 1119.81096 |
| [80] | Anderson, Greg W. and Guionnet, Alice and Zeitouni, Ofer, An introduction to random matrices, Cambridge Studies in Advanced Mathematics, 118, xiv+492, (2010), Cambridge University Press, Cambridge · Zbl 1184.15023 |
| [81] | Guionnet, A. and Zeitouni, O., Concentration of the spectral measure for large matrices, Electronic Communications in Probability, 5, 119-136, (2000) · Zbl 0969.15010 · doi:10.1214/ECP.v5-1026 |
| [82] | Gurau, Razvan, Lost in translation: topological singularities in group field theory, Classical and Quantum Gravity, 27, 23, 235023, 20 pages, (2010) · Zbl 1205.83022 · doi:10.1088/0264-9381/27/23/235023 |
| [83] | Gurau, Razvan, Topological graph polynomials in colored group field theory, Annales Henri Poincar\'e. A Journal of Theoretical and Mathematical Physics, 11, 4, 565-584, (2010) · Zbl 1208.81153 · doi:10.1007/s00023-010-0035-6 |
| [84] | Gurau, Razvan, A generalization of the {V}irasoro algebra to arbitrary dimensions, Nuclear Physics. B, 852, 3, 592-614, (2011) · Zbl 1229.81129 · doi:10.1016/j.nuclphysb.2011.07.009 |
| [85] | Gurau, Razvan, Colored group field theory, Communications in Mathematical Physics, 304, 1, 69-93, (2011) · Zbl 1214.81170 · doi:10.1007/s00220-011-1226-9 |
| [86] | Gurau, Razvan, The {\(1/N\)} expansion of colored tensor models, Annales Henri Poincar\'e. A Journal of Theoretical and Mathematical Physics, 12, 5, 829-847, (2011) · Zbl 1218.81088 · doi:10.1007/s00023-011-0101-8 |
| [87] | Razvan Gurau, Double scaling limit in arbitrary dimensions: a toy model, Physical Review D. Particles, Fields, Gravitation, and Cosmology, 84, 12, 124051, 11 pages, (2011) · doi:10.1103/PhysRevD.84.124051 |
| [88] | Gurau, Razvan, The complete {\(1/N\)} expansion of colored tensor models in arbitrary dimension, Annales Henri Poincar\'e. A Journal of Theoretical and Mathematical Physics, 13, 3, 399-423, (2012) · Zbl 1245.81118 · doi:10.1007/s00023-011-0118-z |
| [89] | Gurau, Razvan, The {S}chwinger–{D}yson equations and the algebra of constraints of random tensor models at all orders, Nuclear Physics. B, 865, 1, 133-147, (2012) · Zbl 1262.81138 · doi:10.1016/j.nuclphysb.2012.07.028 |
| [90] | Gurau, Razvan, The {\(1/N\)} expansion of tensor models beyond perturbation theory, Communications in Mathematical Physics, 330, 3, 973-1019, (2014) · Zbl 1297.81126 · doi:10.1007/s00220-014-1907-2 |
| [91] | Gurau, Razvan, Universality for random tensors, Annales de l’Institut Henri Poincar\'e Probabilit\'es et Statistiques, 50, 4, 1474-1525, (2014) · Zbl 1318.60010 · doi:10.1214/13-AIHP567 |
| [92] | Gurau, Razvan, Random tensors, (2016), Oxford University Press, Oxford · Zbl 1346.83030 |
| [93] | Gurau, Razvan G. and Krajewski, Thomas, Analyticity results for the cumulants in a random matrix model, Annales de l’Institut Henri Poincar\'e D. Combinatorics, Physics and their Interactions, 2, 2, 169-228, (2015) · Zbl 1353.60009 · doi:10.4171/AIHPD/17 |
| [94] | Gurau, Razvan and Magnen, Jacques and Rivasseau, Vincent and Vignes-Tourneret, Fabien, Renormalization of non-commutative {\( \Phi^4_4\)} field theory in {\(x\)} space, Communications in Mathematical Physics, 267, 2, 515-542, (2006) · Zbl 1113.81101 · doi:10.1007/s00220-006-0055-8 |
| [95] | Gurau, Razvan and Rivasseau, V., The \(1/N\) expansion of colored tensor models in arbitrary dimension, Europhysics Letters, 95, 5, 50004, 5 pages, (2011) · doi:10.1209/0295-5075/95/50004 |
| [96] | Gurau, Razvan and Rivasseau, Vincent, The multiscale loop vertex expansion, Annales Henri Poincar\'e. A Journal of Theoretical and Mathematical Physics, 16, 8, 1869-1897, (2015) · Zbl 1321.81046 · doi:10.1007/s00023-014-0370-0 |
| [97] | Gurau, Razvan and Ryan, James P., Colored tensor models – a review, SIGMA. Symmetry, Integrability and Geometry. Methods and Applications, 8, 020, 78 pages, (2012) · Zbl 1242.05094 · doi:10.3842/SIGMA.2012.020 |
| [98] | Gurau, Razvan and Ryan, James P., Melons are branched polymers, Annales Henri Poincar\'e. A Journal of Theoretical and Mathematical Physics, 15, 11, 2085-2131, (2014) · Zbl 1303.83012 · doi:10.1007/s00023-013-0291-3 |
| [99] | Gurau, Razvan and Schaeffer, Gilles, Regular colored graphs of positive degree, (None) · Zbl 1352.05090 |
| [100] | Gurau, Razvan and Tanasa, Adrian and Youmans, Donald R., The double scaling limit of the multi-orientable tensor model, Europhysics Letters, 111, 2, 21002, 6 pages, (2015) · doi:10.1209/0295-5075/111/21002 |
| [101] | Kazakov, V. A., Bilocal regularization of models of random surfaces, Physics Letters B, 150, 4, 282-284, (1985) · doi:10.1016/0370-2693(85)91011-1 |
| [102] | Kazakov, V. A., Ising model on a dynamical planar random lattice: exact solution, Physics Letters. A, 119, 3, 140-144, (1986) · doi:10.1016/0375-9601(86)90433-0 |
| [103] | Kazakov, V. A., The appearance of matter fields from quantum fluctuations of {\(2\)}{D}-gravity, Modern Physics Letters A. Particles and Fields, Gravitation, Cosmology, Nuclear Physics, 4, 22, 2125-2139, (1989) · doi:10.1142/S0217732389002392 |
| [104] | Kegeles, A. and Oriti, D., Continuous point symmetries in group field theories, (None) · Zbl 1362.81066 |
| [105] | Knizhnik, V. G. and Polyakov, A. M. and Zamolodchikov, A. B., Fractal structure of {\(2\)}{D}-quantum gravity, Modern Physics Letters A. Particles and Fields, Gravitation, Cosmology, Nuclear Physics, 3, 8, 819-826, (1988) · doi:10.1142/S0217732388000982 |
| [106] | Krajewski, Thomas, Schwinger–{D}yson equations in group field theories of quantum gravity, Symmetries and Groups in Contemporary Physics, Nankai Ser. Pure Appl. Math. Theoret. Phys., 11, 373-378, (2013), World Sci. Publ., Hackensack, NJ · Zbl 1298.83055 · doi:10.1142/9789814518550_0050 |
| [107] | Lahoche, Vincent and Oriti, Daniele, Renormalization of a tensorial field theory on the homogeneous space {\({\rm SU}(2)/{\rm U}(1)\)}, (None) · Zbl 1357.81145 |
| [108] | Lahoche, Vincent and Samary, Dine Ousmane, Functional renormalisation group for the {\(U(1)-T_5^6\)} TGFT with closure constraint, (None) · Zbl 1418.81056 |
| [109] | Le Gall, Jean-Fran{\c{c}}ois, The topological structure of scaling limits of large planar maps, Inventiones Mathematicae, 169, 3, 621-670, (2007) · Zbl 1132.60013 · doi:10.1007/s00222-007-0059-9 |
| [110] | Le Gall, Jean-Fran{\c{c}}ois, Geodesics in large planar maps and in the {B}rownian map, Acta Mathematica, 205, 2, 287-360, (2010) · Zbl 1214.53036 · doi:10.1007/s11511-010-0056-5 |
| [111] | Le Gall, Jean-Fran{\c{c}}ois, Uniqueness and universality of the {B}rownian map, The Annals of Probability, 41, 4, 2880-2960, (2013) · Zbl 1282.60014 · doi:10.1214/12-AOP792 |
| [112] | Magnen, Jacques and Rivasseau, Vincent, Constructive {\( \phi^4\)} field theory without tears, Annales Henri Poincar\'e. A Journal of Theoretical and Mathematical Physics, 9, 2, 403-424, (2008) · Zbl 1141.81022 · doi:10.1007/s00023-008-0360-1 |
| [113] | Makeenko, Yuri, Loop equations and Virasoro constraints in matrix models, (None) · Zbl 0925.81003 |
| [114] | Marchal, O. and Eynard, B. and Berg{\`e}re, M., The sine-law gap probability, {P}ainlev\'e 5, and asymptotic expansion by the topological recursion, Random Matrices. Theory and Applications, 3, 3, 1450013, 41 pages, (2014) · Zbl 1314.15027 · doi:10.1142/S2010326314500130 |
| [115] | Mehta, Madan Lal, Random matrices, Pure and Applied Mathematics (Amsterdam), 142, xviii+688, (2004), Elsevier/Academic Press, Amsterdam · Zbl 1107.15019 |
| [116] | Ooguri, Hirosi, Topological lattice models in four dimensions, Modern Physics Letters A. Particles and Fields, Gravitation, Cosmology, Nuclear Physics, 7, 30, 2799-2810, (1992) · Zbl 0968.57501 · doi:10.1142/S0217732392004171 |
| [117] | Oriti, Daniele, The microscopic dynamics of quantum space as a group field theory, Foundations of Space and Time, 257-320, (2012), Cambridge University Press, Cambridge · Zbl 1269.83008 |
| [118] | Oriti, Daniele, Group field theory and loop quantum gravity, (None) · Zbl 1459.83019 |
| [119] | Oriti, Daniele and Pranzetti, Daniele and Ryan, James P. and Sindoni, Lorenzo, Generalized quantum gravity condensates for homogeneous geometries and cosmology, Classical and Quantum Gravity, 32, 23, 235016, 40 pages, (2015) · Zbl 1329.83232 · doi:10.1088/0264-9381/32/23/235016 |
| [120] | Oriti, Daniele and Pranzetti, Daniele and Sindoni, Lorenzo, Horizon entropy from quantum gravity condensates, Physical Review Letters, 116, 21, 211301, 6 pages, (2016) · doi:10.1103/PhysRevLett.116.211301 |
| [121] | Oriti, Daniele and Ryan, James P. and Th{\"u}rigen, Johannes, Group field theories for all loop quantum gravity, New Journal of Physics, 17, February, 023042, 46 pages, (2015) · Zbl 1452.83007 · doi:10.1088/1367-2630/17/2/023042 |
| [122] | Oriti, Daniele and Sindoni, Lorenzo and Wilson-Ewing, Edwar , Bouncing cosmologies from quantum gravity condensates, (None) · Zbl 1358.83103 |
| [123] | Oriti, Daniele and Sindoni, Lorenzo and Wilson-Ewing, Edward, Emergent {F}riedmann dynamics with a quantum bounce from quantum gravity condensates, (None) · Zbl 1351.83073 |
| [124] | Politzer, H. David, Reliable perturbative results for strong interactions?, Physical Review Letters, 30, 26, 1346-1349, (1973) · doi:10.1103/PhysRevLett.30.1346 |
| [125] | Rivasseau, Vincent, Constructive matrix theory, Journal of High Energy Physics. A SISSA Journal, 2007, 9, no. 9, 008, 13 pages, (2007) · doi:10.1088/1126-6708/2007/09/008 |
| [126] | Rivasseau, V., Constructive field theory in zero dimension, Advances in Mathematical Physics, 2009, 180159, 12 pages, (2009) · Zbl 1201.81085 · doi:10.1155/2009/180159 |
| [127] | Rivasseau, Vincent, Quantum gravity and renormalization: the tensor track, AIP Conference Proceedings, 1444, 18-29, (2012) · doi:10.1063/1.4715396 |
| [128] | Rivasseau, Vincent, The tensor track: an update, Symmetries and Groups in Contemporary Physics, Nankai Ser. Pure Appl. Math. Theoret. Phys., 11, 63-74, (2013), World Sci. Publ., Hackensack, NJ · Zbl 1298.83059 · doi:10.1142/9789814518550_0011 |
| [129] | Rivasseau, Vincent, The tensor theory space, Fortschritte der Physik. Progress of Physics, 62, 9-10, 835-840, (2014) · Zbl 1338.83086 · doi:10.1002/prop.201400057 |
| [130] | Rivasseau, Vincent, The tensor track, {III}, Fortschritte der Physik. Progress of Physics, 62, 2, 81-107, (2014) · Zbl 1338.83085 · doi:10.1002/prop.201300032 |
| [131] | Ryan, James P., Tensor models and embedded Riemann surfaces, Physical Review D. Particles, Fields, Gravitation, and Cosmology, 85, 2, 024010, 9 pages, (2012) · doi:10.1103/PhysRevD.85.024010 |
| [132] | Salam, A., Weak and electromagnetic interactions, Elementary Particle Theory, 367-377, (1968), Wiley, New York, Almqvist and Wiksell, Stockholm |
| [133] | Samary, Dine Ousmane, Beta functions of \({\rm U}(1)^d\) gauge invariant just renormalizable tensor models, Physical Review D. Particles, Fields, Gravitation, and Cosmology, 88, 10, 105003, 15 pages, (2013) · doi:10.1103/PhysRevD.88.105003 |
| [134] | Samary, Dine Ousmane, Closed equations of the two-point functions for tensorial group field theory, Classical and Quantum Gravity, 31, 18, 185005, 29 pages, (2014) · Zbl 1300.81058 · doi:10.1088/0264-9381/31/18/185005 |
| [135] | Samary, Dine Ousmane and Vignes-Tourneret, Fabien, Just renormalizable {TGFT}’s on {\({\rm U}(1)^d\)} with gauge invariance, Communications in Mathematical Physics, 329, 2, 545-578, (2014) · Zbl 1294.83031 · doi:10.1007/s00220-014-1930-3 |
| [136] | Sasakura, Naoki, Tensor model for gravity and orientability of manifold, Modern Physics Letters A. Particles and Fields, Gravitation, Cosmology, Nuclear Physics, 6, 28, 2613-2623, (1991) · Zbl 1020.83542 · doi:10.1142/S0217732391003055 |
| [137] | Sasakura, Naoki, Super tensor models, super fuzzy spaces and super \(n\)-ary transformations, International Journal of Modern Physics A. Particles and Fields. Gravitation. Cosmology, 26, 24, 4203-4216, (2011) · Zbl 1247.83058 · doi:10.1142/S0217751X11054449 |
| [138] | Sasakura, Naoki, Tensor models and hierarchy of {\(n\)}-ary algebras, International Journal of Modern Physics A. Particles and Fields. Gravitation. Cosmology, 26, 19, 3249-3258, (2011) · Zbl 1247.83057 · doi:10.1142/S0217751X1105381X |
| [139] | Schaeffer, Gilles, Bijective census and random generation of {E}ulerian planar maps with prescribed vertex degrees, Electronic Journal of Combinatorics, 4, 1, 20, 14 pages, (1997) · Zbl 0885.05076 |
| [140] | Sindoni, Lorenzo, Effective equations for {GFT} condensates from fidelity, (None) · Zbl 1242.83046 |
| [141] | ’t Hooft, G., A planar diagram theory for strong interactions, Nuclear Physics. B, 72, 3, 461-473, (1974) · doi:10.1016/0550-3213(74)90154-0 |
| [142] | ’t Hooft, G. and Veltman, M., Regularization and renormalization of gauge fields, Nuclear Physics. B, 44, 1, 189-213, (1972) · doi:10.1016/0550-3213(72)90279-9 |
| [143] | ’t Hooft, G. and Veltman, M., One-loop divergencies in the theory of gravitation, 20, 69-94, (1974) · Zbl 1422.83019 |
| [144] | Tanasa, Adrian, Multi-orientable group field theory, Journal of Physics. A. Mathematical and Theoretical, 45, 16, 165401, 19 pages, (2012) · Zbl 1246.81172 · doi:10.1088/1751-8113/45/16/165401 |
| [145] | Weinberg, S., A model of leptons, Physical Review Letters, 19, 21, 1264-1266, (1967) · doi:10.1103/PhysRevLett.19.1264 |
| [146] | Wigner, Eugene P., Characteristic vectors of bordered matrices with infinite dimensions, Annals of Mathematics. Second Series, 62, 548-564, (1955) · Zbl 0067.08403 · doi:10.2307/1970079 |
| [147] | Wishart, John, The generalised product moment distribution in samples from a normal multivariate population, Biometrika, 20A, 1-2, 32-52, (1928) · JFM 54.0565.02 · doi:10.1093/biomet/20A.1-2.32 |
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