Tensor models are natural generalizations of matrix models. The interactions and observables in the case of unitary invariant models are generalizations of matrix traces. Some notable interactions in the literature include the melonic ones, the tetrahedral one as well as the planar ones in rank three, or necklaces in even ranks. Here we introduce generalized melonic interactions which generalize the melonic and necklace interactions. We characterize them as tree-like gluings of quartic interactions. We also completely characterize the Feynman graphs which contribute to the large $N$ limit. For a subclass of generalized melonic interactions called totally unbalanced interactions, we prove that the large $N$ limit is Gaussian and therefore the Feynman graphs are in bijection with trees. This result further extends the class of tensor models which fall into the Gaussian universality class. Another key aspect of tensor models with generalized melonic interactions is that they can be written as matrix models without increasing the number of degrees of freedom of the original tensor models. In the case of totally unbalanced interactions, this new matrix model formulation in fact decreases the number of degrees of freedom, meaning that some of the original degrees of freedom are effectively integrated. We then show how the large $N$ Gaussian behavior can be reproduced using a saddle point analysis on those matrix models.
Tensor models are natural generalizations of matrix models. The interactions and observables in the case of unitary invariant models are generalizations of matrix traces. Some notable interactions in the literature include the melonic ones, the tetrahedral one as well as the planar ones in rank three, or necklaces in even ranks. Here we introduce generalized melonic interactions which generalize the melonic and necklace interactions. We characterize them as tree-like gluings of quartic interactions. We also completely characterize the Feynman graphs which contribute to the large N limit. For a subclass of generalized melonic interactions called totally unbalanced interactions, we prove that the large N limit is Gaussian and therefore the Feynman graphs are in bijection with trees. This result further extends the class of tensor models which fall into the Gaussian universality class. Another key aspect of tensor models with generalized melonic interactions is that they can be written as matrix models without increasing the number of degrees of freedom of the original tensor models. In the case of totally unbalanced interactions, this new matrix model formulation in fact decreases the number of degrees of freedom, meaning that some of the original degrees of freedom are effectively integrated. We then show how the large N Gaussian behavior can be reproduced using a saddle point analysis on those matrix models.
This thesis focuses on renormalization of tensor field theories. Its first part considers a quartic tensor model with O(N)3 symmetry and long-range propagator. The existence of a non-perturbative fixed point in any d at large N is established. We found four lines of fixed points parametrized by the so-called tetrahedral coupling. One of them is infrared attractive, strongly interacting and gives rise to a new kind of CFT, called melonic CFTs which are then studied in more details. We first compute dimensions of bilinears and OPE coefficients at the fixed point which are consistent with a unitary CFT at large N. We then compute 1/N corrections. At next-to-leading order, the line of fixed points collapses to one fixed point. However, the corrections are complex and unitarity is broken at NLO. Finally, we show that this model respects the F-theorem. The next part of the thesis investigates sextic tensor field theories in rank 3 and 5. In rank 3, we found two IR stable real fixed points in short range and a line of IR stable real fixed points in long range. Surprisingly, the only fixed point in rank 5 is the Gaussian one. For the rank 3 model, in the short-range case, we still find two IR stable fixed points at NLO. However, in the long-range case, the corrections to the fixed points are non-perturbative and hence unreliable: we found no precursor of the large N fixed point. The last part of the thesis investigates the class of model exhibiting a melonic large N limit. We prove that models with tensors in an irreducible representation of O(N) or Sp(N) in rank 5 indeed admit a large N limit. This generalization relies on recursive bounds derived from a detailed combinatorial analysis of Feynman graphs involved in the perturbative expansion of our model.
Two direct connections between random tensors and random matrices are discussed in this article. In the rst part, we introduce U( \tau ) matrix models which generate fully packed, oriented loops on random surfaces. e latter are found to be in bijection with a set of regular edge-colored graphs. It is shown that the expansion in the number of loops is organized like the 1/ N expansion of rank-three tensor models. Recent results on tensor models are applied in this context. For example, congurations which maximize the number of loops are precisely the melonic graphs of tensor models and a scaling limit which projects onto themelonic sector is found. is approach is generalized to higher-rank tensor models, which generate loops with fugacity \tau on triangulations in dimension d–1 . In the second part, we introduce singular value decompositions to evaluate the expectations of polynomial observables of Gaussian random tensors. Performing the integrals over the unitary group leads to a notion of eective observables which expand onto regular trace invariants. We show that both asymptotic and exact new calculations of expectations can be performed this way.
We construct a measure that exhibits two aspects of a new type of universality and dramatically simplifies the integration of tensors Ta1,a2,…,aD∈C (a1,…,aD=1,…,N) at large N. In contrast to matrix integration, in which matrix traces canonically yield the integrand, tensors need additional information (equivalent to a D-coloured graph B) to contract their indices and form a tensor trace B(T). We show that, whenever each B1,…,Bn can be obtained by a recursive construction known as melonicity, then the leading order in N of the integral of $ {B_1}(T) {B_2}(T) \cdots {B_n}(T) $ is independent of the -- often intricate -- combinatorics of the traces Bi, but also, to our surprise, independent of D as far as D≥3. Instead, at large N, these integrals are some functions (indexed by n) of the number of vertices 2pi of Bi which we call melonic polynomials. Melonic traces cumulants with respect to any ('interacting') measure exp{−ND−1∑i=1mgiBi(T)}dμ0(T)(g1,…,gm∈R,dμ0(T)=the tensor Gaussian) with each Bi melonic, can be computed with our universal measure that replaces each Bi by a canonical trace depending only on pi. We prove that any two melonic tensor models are indistinguishable at large-N, independently of the number of tensor indices (first universality aspect), and of the fine-grainedness of their interactions (second universality), being a sufficient condition that the couplings (the parameters gi above) agree and their respective traces are monomials with the same degree in T.
Abstract We demonstrate that random tensors transforming under rank-5 irreducible representations of O(N) O ( N ) can support melonic large N expansions. Our construction is based on models with sextic (5-simplex) interaction, which generalize previously studied rank-3 models with quartic (tetrahedral) interaction (Benedetti et al. in Commun Math Phys 371:55, 2019. arXiv:1712.00249 ; Carrozza in JHEP 06:039, 2018. arXiv:1803.02496 ). Beyond the irreducible character of the representations, our proof relies on recursive bounds derived from a detailed combinatorial analysis of the Feynman graphs. Our results provide further evidence that the melonic limit is a universal feature of irreducible tensor models in arbitrary rank.
Another proof of the 1/N expansion of the rank three tensor model with tetrahedral interaction
The rank three tensor model with tetrahedral interaction was shown by Carrozza and Tanasa to admit a 1/N expansion, dominated by melonic diagrams, and double tadpoles decorated with melons at next-to-leading order. This model has generated a renewed interest in tensor models because it has the same large N limit as the SYK model. In contrast with matrix models, there is no method which would be able to prove the existence of 1/N expansions in arbitrary tensor models. The method used by Carrozza and Tanasa proves the existence of the 1/N expansion using two-dimensional topology, before identifying the leading order and next-to-leading graphs. However, another method was required for complex, rank three tensor models with planar interactions, which is based on flips. The latter are moves which cut two propagators of Feynman graphs and reglue them differently. They allow transforming graphs while tracking their orders in the 1/N expansion. Here we use this method to re-prove the results of Carrozza and Tanasa, thereby proving the existence of the 1/N expansion, the melonic dominance at leading order and the melon-decorated double tadpoles at next-to-leading order, all in one go.
The first part of these lecture notes is mostly devoted to a comparative discussion of the three basic large N limits, which apply to fields which are vectors, matrices, or tensors of rank three and higher. After a brief review of some physical applications of large N limits, we present a few solvable examples in zero space-time dimension. Using models with fields in the fundamental representation of O(N), O(N)2, or O(N)3 symmetry, we compare their combinatorial properties and highlight a competition between the snail and melon diagrams. We exhibit the different methods used for solving the vector, matrix, and tensor large N limits. In the latter example we review how the dominance of melonic diagrams follows when a special ``tetrahedral" interaction is introduced. The second part of the lectures is mostly about the fermionic quantum mechanical tensor models, whose large N limits are similar to that in the Sachdev-Ye-Kitaev (SYK) model. The minimal Majorana model with O(N)3 symmetry and the tetrahedral Hamiltonian is reviewed in some detail; it is the closest tensor counterpart of the SYK model. Also reviewed are generalizations to complex fermionic tensors, including a model with SU(N)2×O(N)×U(1) symmetry, which is a tensor counterpart of the complex SYK model. The bosonic large N tensor models, which are formally tractable in continuous spacetime dimension, are reviewed briefly at the end.