Critical behavior of colored tensor models in the large N limit

By Valentin Bonzom, R. Gurau, A. Riello, Vincent Rivasseau. 2011

362 citations recorded by the source corpus.

Colored tensor models have been recently shown to admit a large N expansion, whose leading order encodes a sum over a class of colored triangulations of the D-sphere. The present paper investigates in details this leading order. We show that the relevant triangula- tions proliferate like a species of colored trees. The leading order is therefore summable and exhibits a critical behavior, independent of the dimension. A continuum limit is reached by tuning the coupling constant to its critical value while inserting an infinite number of pairs of D-simplices glued together in a specific way. We argue that the dominant triangulations are branched polymers. Tensor models (1-3) and group field theories (4-7) are the natural generalization of matrix models (8, 9) implementing in a consistent way the sum over random triangulations in dimensions higher than two. They are notoriously hard to control analytically and one usually resorts to numerical simulations (10-12). Progress has recently been made in the analytic control of tensor models with the advent of the 1/N expansion (13-15) of colored (16-18) tensor models. This expansion synthetizes several alternative evaluations of graph amplitudes in tensor models (19-27) and provides a straightforward generalization of the familiar genus expansion of matrix models (28, 29) in arbitrary dimension. The coloring of the fields allows one to address previously inaccessible questions in tensor models like the implementation of the diffeomorphism symmetry (27, 30) in the Boulatov model or the identification of embedded matrix models (31). The symmetries of tensor models have recently been studied using n-ary algebras (32, 33). This paper is the first in a long series of studies of the implications of the 1/N expansion in colored tensor models. We present here a complete analysis of the leading order in the large N limit in arbitrary dimensions, indexed by graphs of spherical topology (14). To perform the study of this leading order one needs to address the following two questions • What is the combinatorics of the Feynman graphs contributing to the leading order, i.e. the higher dimensional extension of the notion of planar graphs? Unlike in matrix models, where planarity and spherical topology are trivially related, this question is non trivial in tensor models. In particular not all triangulations of the sphere contribute to the leading order. • Is the series of the leading order summable with a non zero radius of convergence? If this is the case, then, in the large N limit, the model exhibits a critical behavior whose critical exponents one needs to compute.

Subfields: Computational Mathematics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/481aab15783f0ba9668a17119a449bc534614f3a

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Citations in the corpus

A Renormalizable 4-Dimensional Tensor Field Theory (2011, background): “Results on statistical mechanics on random D ≥ 3 geometries followed quickly [41, 42, 43].”
All point correlation functions in SYK (2017, background): “nd so on. The diagrams contributing to the two-point function consist entirely of melons. This is true of SYK, as well as variants of SYK [11,13{17] and their extensions, and of certain tensor models [4,18,12,19{22] and their extensions. A cut of a melon diagram gives a ladder diagram, contributing ”
Universality for Random Tensors (2011, methodology): “They have been extensively discussed in [5].”
The Spin-Foam Approach to Quantum Gravity (2012, methodology): “In a further modification of the previous model (known as coloured tensor models [202]) new techniques based on a suitable 1/N expansion imply that amplitudes are dominated by spherical topology [201]; moreover, it seem possible that the continuum limit might be critical as in certain matrix models ”
Uncolored random tensors, melon diagrams, and the Sachdev-Ye-Kitaev models (2016, background): “If they are not empty one can argue (see [12]) that they are also maximal graphs.”
The complete 1/N expansion of a SYK–like tensor model (2016, background): “Two equivalent definitions of melonic graphs can be found in the literature [2, 12], and we present and use them both.”
O(N) Random Tensor Models (2015, methodology): “This is achieved through the use of various analytic combinatorics techniques.”
An SYK-like model without disorder (2016, methodology): “In section 3, following [22], we sketch the proof that the Feynman diagrams that survive in the large N limit of the tensor model are precisely those of the SYK model.”
Random tensor models in the large N limit: Uncoloring the colored tensor models (2012, background): “These graphs, baptized melonic [41], again proliferate only exponentially, as they map to colored (D+1)-ary trees [41, 42].”
Renormalization of Tensorial Group Field Theories: Abelian U(1) Models in Four Dimensions (2012, methodology): “This 1/N expansion has allowed the first proofs that a phase transition indeed occurs in simple tensor models [76–78], and is a key tool in the analysis we perform in this paper.”