We study the double scaling limit of the $O(N)^3$-invariant tensor model, initially introduced in Carrozza and Tanasa, Lett. Math. Phys. (2016). This model has an interacting part containing two types of quartic invariants, the tetrahedric and the pi
llow one. For the 2-point function, we rewrite the sum over Feynman graphs at each order in the $1/N$ expansion as a emph{finite} sum, where the summand is a function of the generating series of melons and chains (a.k.a. ladders). The graphs which are the most singular in the continuum limit are characterized at each order in the $1/N$ expansion. This leads to a double scaling limit which picks up contributions from all orders in the $1/N$ expansion. In contrast with matrix models, but similarly to previous double scaling limits in tensor models, this double scaling limit is summable. The tools used in order to prove our results are combinatorial, namely a thorough diagrammatic analysis of Feynman graphs, as well as an analysis of the singularities of the relevant generating series.
We study a $b$-deformation of monotone Hurwitz numbers, obtained by deforming Schur functions into Jack symmetric functions. It is a special case of the $b$-deformed weighted Hurwitz numbers recently introduced by the last two authors and has an inte
rpretation in terms of generalized branched coverings of the sphere by non-oriented surfaces. We give an evolution (cut-and-join) equation for this model and we derive, by a method of independent interest, explicit Virasoro constraints from it, for arbitrary values of the deformation parameter $b$. We apply them to prove a conjecture of Feray on Jack characters. We also provide a combinatorial model of non-oriented monotone Hurwitz maps, which generalizes monotone transposition factorizations. In the case $b=1$ we show that the model obeys the BKP hierarchy of Kac and Van de Leur. As a consequence of our analysis we prove a recent conjecture of Oliveira and Novaes relating zonal polynomials with the dimensions of irreducible representations of $O(N)$. We also relate the model to an $O(N)$ version of the Brezin-Gross-Witten integral, which we solve explicitly in terms of Pfaffians in the case of even multiplicities.
Tensor models are generalizations of matrix models and as such, it is a natural question to ask whether they satisfy some form of the topological recursion. The world of unitary-invariant observables is however much richer in tensor models than in ma
trix models. It is therefore a priori unclear which set of observables could satisfy the topological recursion. Such a set of observables was identified a few years ago in the context of the quartic melonic model by the first author and Dartois. It was shown to satisfy an extension of the topological recursion introduced by Borot and called the blobbed topological recursion. Here we show that this set of observables is present in arbitrary tensor models which have non-vanishing couplings for the quartic melonic interactions. It satisfies the blobbed topological recursion in a universal way, i.e. independently of the choices of the other interactions. In combinatorial terms, the correlation functions describe stuffed maps with colored boundary components. The specifics of the model only appear in the generating functions of the stuffings and the blobbed topological recursion only requires them to have well-defined $1/N$ expansions. The spectral curve is a disjoint union of Gaussian spectral curves, with the cylinder function receiving an additional holomorphic part. This result is achieved via a perturbative rewriting of tensor models as multi-matrix models due to the first author, Lionni and Rivasseau. It is then possible to formally integrate all degrees of freedom except those which enter the topological recursion, meaning interpreting the Feynman graphs as stuffed maps. We further provide new expressions to relate the expectations of $U(N)^d$-invariant observables on the tensor and matrix sides.
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 renewe
d 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.
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 tetra
hedral 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.
The relation between the 2d Ising partition function and spin network evaluations, reflecting a bulk-boundary duality between the 2d Ising model and 3d quantum gravity, promises an exchange of results and methods between statistical physics and quant
um geometry. We apply this relation to the case of the tetrahedral graph. First, we find that the high/low temperature duality of the 2d Ising model translates into a new self-duality formula for Wigners 6j-symbol from the theory of spin recoupling. Second, we focus on the duality between the large spin asymptotics of the 6j-symbol and Fisher zeros. Using the Ponzano-Regge formula for the asymptotics for the 6j-symbol at large spins in terms of the tetrahedron geometry, we obtain a geometric formula for the zeros of the (inhomogeneous) Ising partition function in terms of triangle angles and dihedral angles in the tetrahedron. While it is well-known that the 2d intrinsic geometry can be used to parametrize the critical point of the Ising model, e.g. on isoradial graphs, it is the first time to our knowledge that the extrinsic geometry is found to also be relevant.This outlines a method towards a more general geometric parametrization of the Fisher zeros for the 2d Ising model on arbitrary graphs.
A crucial result on the celebrated Sachdev-Ye-Kitaev model is that its large $N$ limit is dominated by melonic graphs. In this letter we offer a rigorous, diagrammatic proof of that result by direct, combinatorial analysis of its Feynman graphs.
Colored triangulations offer a generalization of combinatorial maps to higher dimensions. Just like maps are gluings of polygons, colored triangulations are built as gluings of special, higher-dimensional building blocks, such as octahedra, which we
call colored building blocks and known in the dual as bubbles. A colored building block is determined by its boundary triangulation, which in the case of polygons is simply characterized by its length. In three dimensions, colored building blocks are labeled by some 2D triangulations and those homeomorphic to the 3-ball are labeled by the subset of planar ones. Similarly to Eulers formula in 2D which provides an upper bound to the number of vertices at fixed number of polygons with given lengths, we look in three dimensions for an upper bound on the number of edges at fixed number of given colored building blocks. In this article we solve this problem when all colored building blocks, except possibly one, are homeomorphic to the 3-ball. To do this, we find a characterization of the way a colored building block homeomorphic to the ball has to be glued to other blocks of arbitrary topology in a colored triangulation which maximizes the number of edges. This local characterization can be extended to the whole triangulation as long as there is at most one colored building block which is not a 3-ball. The triangulations obtained this way are in bijection with trees. The number of edges is given as an independent sum over the building blocks of such a triangulation. In the case of all colored building blocks being homeomorphic to the 3-ball, we show that these triangulations are homeomorphic to the 3-sphere. Those results were only known for the octahedron and for melonic building blocks before. This article is self-contained and can be used as an introduction to colored triangulations and their bubbles from a purely combinatorial point of view.
The Sachdev-Ye-Kitaev (SYK) model is a model of $q$ interacting fermions. Gross and Rosenhaus have proposed a generalization of the SYK model which involves fermions with different flavors. In terms of Feynman graphs, those flavors are reminiscent of
the colors used in random tensor theory. This gives us the opportunity to apply some modern, yet elementary, tools developed in the context of random tensors to one particular instance of such colored SYK models. We illustrate our method by identifying all diagrams which contribute to the leading and next-to-leading orders of the 2-point and 4-point functions in the large $N$ expansion, and argue that our method can be further applied if necessary. In a second part we focus on the recently introduced Gurau-Witten tensor model and also extract the leading and next-to-leading orders of the 2-point and 4-point functions. This analysis turns out to be remarkably more involved than in the colored SYK model.
Random tensor models are generalizations of random matrix models which admit $1/N$ expansions. In this article we show that the topological recursion, a modern approach to matrix models which solves the loop equations at all orders, is also satisfied
in some tensor models. While it is obvious in some tensor models which are matrix models in disguise, it is far from clear that it can be applied to others. Here we focus on melonic interactions for which the models are best understood, and further restrict to the quartic case. Then Hubbard-Stratonovich transformation maps the tensor model to a multi-matrix model with multi-trace interactions. We study this matrix model and show that after substracting the leading order, it satisfies the blobbed topological recursion. It is a new extension of the topological recursion, recently introduced by Borot and further studied by Borot and Shadrin. Here it applies straightforwardly, yet with a novelty as our model displays a disconnected spectral curve, which is the union of several spectral curves of the Gaussian Unitary Ensemble. Finally, we propose a way to evaluate expectations of tensorial observables using the correlation functions computed from the blobbed topological recursion.
