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arXiv:2607.28518

Redundant moments of non-melonic random tensor models: simplifying the tensor bootstrap

Carlos I. Pérez-Sánchez

Published Jul 30, 2026• 0 citations
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Abstract

We prove the redundancy of a family (the so-called melonic family) of large-$N$ moments in arbitrary tensor models. This reduces the number of independent entries in the positive semidefinite matrices that build the core of the tensor bootstrap (for tensor models there are already several generalizations of the positive semidefinite Toeplitz and Hankel matrices of lattice gauge theory and random matrix bootstrap, respectively). More concretely, we prove that any melonic operator $C$ at large-$N$ has an expectation value that depends on $C$ exclusively through its degree. A similar statement is shown here to hold for the Schwinger-Dyson equations of melonic moments. The strength of our result is also its scope, which is not limited to melonic tensor models.

DOI
10.48550/arxiv.2607.28518

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We prove the redundancy of a family (the so-called melonic family) of large-NNN moments in arbitrary tensor models. This reduces the number of independent entries in the positive semidefinite matrices that build the core of the tensor bootstrap (for tensor models there are already several generalizations of the positive semidefinite Toeplitz and Hankel matrices of lattice gauge theory and random matrix bootstrap, respectively). More concretely, we prove that any melonic operator CCC at large-NNN has an expectation value that depends on CCC exclusively through its degree. A similar statement is shown here to hold for the Schwinger-Dyson equations of melonic moments. The strength of our result is also its scope, which is not limited to melonic tensor models.
This work proposes a bootstrapping with positivity methodology to study random U(N)DU(N)^{D}U(N)D invariant tensors in the large NNN limit. As has been done for U(N)U(N)U(N) invariant random matrices, we combine the Dyson-Schwinger equations and positivity constraints of moments to approximate the moments of such tensor models. As examples, we bootstrap the quartic and two hexic rank three tensor models. All models studied converge quickly, and for those which have known analytic formulae, they converge to such solutions. We conjecture new explicit formulae for all moments of the rank three quartic model and support this conjecture using bootstrapped results and explicit double-series computations with 'feyntensor'.
We construct a measure that exhibits two aspects of a new type of universality and dramatically simplifies the integration of tensors Ta1,a2,…,aD∈CT_{a_1,a_2,\ldots,a_D} \in \mathbb{C}Ta1​,a2​,…,aD​​∈C (a1,…,aD=1,…,Na_1,\ldots,a_D=1,\ldots,Na1​,…,aD​=1,…,N) at large NNN. In contrast to matrix integration, in which matrix traces canonically yield the integrand, tensors need additional information (equivalent to a DDD-coloured graph BBB) to contract their indices and form a tensor trace B(T)B(T)B(T). We show that, whenever each B1,…,BnB_1,\ldots, B_nB1​,…,Bn​ can be obtained by a recursive construction known as melonicity, then the leading order in NNN of the integral of $ {B_1}(T) {B_2}(T) \cdots {B_n}(T) $ is independent of the -- often intricate -- combinatorics of the traces BiB_iBi​, but also, to our surprise, independent of DDD as far as D≥3D\geq 3D≥3. Instead, at large NNN, these integrals are some functions (indexed by nnn) of the number of vertices 2pi2p_i2pi​ of BiB_iBi​ 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) \exp\Big\{-N^{D-1} \sum_{i=1}^m g_i {B_i}(T)\Big\} \mathrm{d}μ_0(T) \quad (g_1,\ldots,g_m \in \mathbb{R}, \mathrm{d}μ_0(T) =\text{the tensor Gaussian}) exp{−ND−1∑i=1m​gi​Bi​(T)}dμ0​(T)(g1​,…,gm​∈R,dμ0​(T)=the tensor Gaussian) with each BiB_iBi​ melonic, can be computed with our universal measure that replaces each BiB_iBi​ by a canonical trace depending only on pip_ipi​. We prove that any two melonic tensor models are indistinguishable at large-NNN, 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 gig_igi​ above) agree and their respective traces are monomials with the same degree in TTT.
Abstract In the first section we give a briefly presentation of the U(N)D-invariant tensor models (N being again the size of the tensor, and D being the dimension). The next section is then dedicated to the analysis of the Dyson–Schwinger equations (DSE) in the large N limit. These results are essential to implement the double scaling limit mechanism of the DSEs, which is done in the third section. The main result of this chapter is the doubly-scaled 2-point function for a model with generic melonic interactions. However, several assumptions on the large N scaling of cumulants are made along the way. They are proved using various combinatorial methods.
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 NNN limit. For a subclass of generalized melonic interactions called totally unbalanced interactions, we prove that the large NNN 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 NNN Gaussian behavior can be reproduced using a saddle point analysis on those matrix models.
Abstract We demonstrate that random tensors transforming under rank-5 irreducible representations of O(N)\mathrm {O}(N)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.
Tensor models and tensor field theories admit a 1/N expansion and a melonic large N limit which is simpler than the planar limit of random matrices and richer than the large N limit of vector models. They provide examples of analytically tractable but non-trivial strongly coupled quantum field theories and lead to a new class of conformal field theories. We present a compact introduction to the topic, covering both some of the classical results in the field, like the details of the 1/N expansion, as well as recent developments. These notes are loosely bases on four lectures given at the Journées de physique mathématique Lyon 2019: Random tensors and SYK models.