arXiv subjects

Carlos I. Perez-Sanchez

Publications and source records attributed to Carlos I. Perez-Sanchez.

12 recordsLinked to original sources

arXiv2026

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

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.

math-ph↗

arXiv2026

Twofold universality of large-$N$ melonic random tensors

We construct a measure that exhibits two aspects of a new type of universality and dramatically simplifies the integration of tensors $T_{a_1,a_2,\ldots,a_D} \in \mathbb{C}$ ($a_1,\ldots,a_D=1,\ldots,N$) 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 $B_1,\ldots, B_n$ 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 $B_i$, but also, to our surprise, independent of $D$ as far as $D\geq 3$. Instead, at large $N$, these integrals are some functions (indexed by $n$) of the number of vertices $2p_i$ of $B_i$ which we call melonic polynomials. Melonic traces cumulants with respect to any ('interacting') measure \[ \exp\Big\{-N^{D-1} \sum_{i=1}^m g_i {B_i}(T)\Big\} \mathrm{d}μ_0(T) \quad (g_1,\ldots,g_m \in \mathbb{R}, \mathrm{d}μ_0(T) =\text{the tensor Gaussian}) \] with each $B_i$ melonic, can be computed with our universal measure that replaces each $B_i$ by a canonical trace depending only on $p_i$. 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 $g_i$ above) agree and their respective traces are monomials with the same degree in $T$.

math.CO↗

arXiv2025

Comment on 'Gauge networks in noncommutative geometry'

The article (Gauge networks in noncommutative geometry, J. Geom. Phys. 75 : 71--91, 2014) that motivates this comment provides, in particular, one answer to the following natural question: what is noncommutative geometry on a lattice? In the context of spectral triples, Marcolli and van Suijlekom define in op. cit. a Dirac operator on the lattice and identify the corresponding Spectral Action with the lattice Yang-Mills--Higgs system. In this comment we show that the continuum limit of this theory is the Yang-Mills action functional, without a Higgs scalar. Résumé: Qu'est-ce que la géométrie non-commutative sur réseau ? À cette question l'article ici commenté (Gauge networks in noncommutative geometry, J. Geom. Phys. 75 : 71--91, 2014) apporte une des réponses possibles. Marcolli et van Suijlekom, travaillant dans le contexte des triplets spectraux, y construisent un opérateur de type Dirac pour le réseau et dérivent une théorie sur réseau de type Yang-Mills--Higgs à partir de l'Action Spectrale. Ce commentaire montre que la limite continue de ce modèle est la théorie pure de Yang-Mills (sans aucun Higgs).

math-ph↗

arXiv2025

Bratteli networks and the Spectral Action on quivers

In the context of noncommutative geometry, we consider quiver representations -- not on vector spaces, as traditional, but on finite-dimensional prespectral triples (`discrete topological noncommutative spaces'). A similar idea appeared in the original work of Marcolli-van Suijlekom on quiver representations in spectral triples (`discrete noncommutative geometries'), which paved the way for some of our results in independent directions. We introduce Bratteli networks, a structure that yields a neat combinatorial characterisation of the space $\mathrm{Rep}~Q$ of prespectral-triple-representations of a quiver $\mathrm{Rep}~Q$, as well as of the gauge group and of their quotient. Not only these claims that make it possible to `integrate over $\mathrm{Rep}~Q$' are, as we now argue, in line with the spirit of random noncommutative geometry -- formulating path integrals over Dirac operators -- but they also contain a physically relevant case. Namely, the equivalence between quiver representations and path algebra modules, established here for the new category, inspired the following construction: Only from representation theory data, we build a spectral triple for the quiver and evaluate the spectral action functional from a general formula over closed paths. When we apply this construction to lattice-quivers, we obtain not only Wilsonian Yang-Mills lattice gauge theory, but also the Weisz-Wohlert-cells in the context of Symanzik's improved gauge theory. We show that a hermitian (`Higgs') matrix field emerges from the self-loops of the quiver and derive the Yang-Mills--Higgs theory on flat space as a smooth limit.

math.RT↗

arXiv2022

Borel summability of the 1/N expansion in quartic O(N)-vector models

We consider a quartic O(N)-vector model. Using the Loop Vertex Expansion, we prove the Borel summability in 1/N along the real axis of the partition function and of the connected correlations of the model. The Borel summability holds uniformly in the coupling constant, as long as the latter belongs to a cardioid like domain of the complex plane, avoiding the negative real axis.

math-ph↗

arXiv2022

A ribbon graph derivation of the algebra of functional renormalization for random multi-matrices with multi-trace interactions

We focus on functional renormalization for ensembles of several (say $n\geq 1$) random matrices, whose potentials include multi-traces, to wit, the probability measure contains factors of the form $ \exp[-\mathrm{Tr}(V_1)\times\ldots\times \mathrm{Tr}(V_k)]$ for certain noncommutative polynomials $V_1,\ldots,V_k\in \mathbb{C}_{\langle n \rangle}$ in the $n$ matrices. This article shows how the "algebra of functional renormalization" -- that is, the structure that makes the renormalization flow equation computable -- is derived from ribbon graphs, only by requiring the one-loop structure that such equation (due to Wetterich) is expected to have. Whenever it is possible to compute the renormalization flow in terms of $\mathrm U(N)$-invariants, the structure gained is the matrix algebra $M_n( \mathcal{A}_{n,N}, \star ) $ with entries in $\mathcal{A}_{n,N}=(\mathbb{C}_{\langle n \rangle} \otimes \mathbb{C}_{\langle n \rangle} )\oplus( \mathbb{C}_{\langle n \rangle} \boxtimes \mathbb{C}_{\langle n \rangle})$, being $\mathbb{C}_{\langle n \rangle} $ the free algebra generated by the $n$ Hermitian matrices of size $N$ (the flowing random variables) with multiplication of homogeneous elements in $\mathcal{A}_{n,N}$ given, for each $P,Q,U,W\in\mathbb{C}_{\langle n \rangle}$, by \begin{align*}(U \otimes W) \star ( P\otimes Q) &= PU \otimes WQ \,, & (U\boxtimes W) \star ( P\otimes Q) &=U \boxtimes PWQ \,, \\(U \otimes W) \star ( P\boxtimes Q) &= WPU \boxtimes Q \,,\ & (U\boxtimes W) \star ( P\boxtimes Q) &= \mathrm{Tr} (WP) U\boxtimes Q \,,\end{align*} which, together with the condition $(λU) \boxtimes W = U\boxtimes (λW) $ for each complex $λ$, fully define the symbol $\boxtimes$.

math-ph↗

arXiv2021

On multimatrix models motivated by random noncommutative geometry II: A Yang-Mills-Higgs matrix model

We continue the study of fuzzy geometries inside Connes' spectral formalism and their relation to multimatrix models. In this companion paper to [arXiv 2007:10914, Ann. Henri Poincaré] we propose a gauge theory setting based on noncommutative geometry, which -- just as the traditional formulation in terms of almost-commutative manifolds -- has the ability to also accommodate a Higgs field. However, in contrast to "almost-commutative manifolds", the present framework employs only finite dimensional algebras which we call gauge matrix spectral triples. In a path-integral quantization approach to the Spectral Action, this allows to state Yang-Mills--Higgs theory (on four-dimensional Euclidean fuzzy space) as an explicit random multimatrix model obtained here, whose matrix fields mirror those of the Yang-Mills--Higgs theory on a smooth manifold.

math-ph↗

arXiv2021

Comment on "The phase diagram of the multi-matrix model with ABAB-interaction from functional renormalization"

Recently, [JHEP 20 131 (2020)] obtained (a similar, scaled version of) the ($a,b$)-phase diagram derived from the Kazakov--Zinn-Justin solution of the Hermitian two-matrix model with interactions \[\mathrm{Tr\,}\Big\{\frac{a}{4} (A^4+B^4)+\frac{b}{2} ABAB\Big\}\,,\] starting from Functional Renormalization. We comment on something unexpected: the phase diagram of [JHEP 20 131 (2020)] is based on a $β_b$-function that does not have the one-loop structure of the Wetterich-Morris Equation. This raises the question of how to reproduce the phase diagram from a set of $β$-functions that is, in its totality, consistent with Functional Renormalization. A non-minimalist, yet simple truncation that could lead to the phase diagram is provided. Additionally, we identify the ensemble for which the result of op. cit. would be entirely correct.

hep-th↗

arXiv2021

On multimatrix models motivated by random Noncommutative Geometry I: the Functional Renormalization Group as a flow in the free algebra

Random noncommutative geometry can be seen as a Euclidean path-integral approach to the quantization of the theory defined by the Spectral Action in noncommutative geometry (NCG). With the aim of investigating phase transitions in random NCG of arbitrary dimension, we study the non-perturbative Functional Renormalization Group for multimatrix models whose action consists of noncommutative polynomials in Hermitian and anti-Hermitian matrices. Such structure is dictated by the Spectral Action for the Dirac operator in Barrett's spectral triple formulation of fuzzy spaces.The present mathematically rigorous treatment puts forward "coordinate-free" language that might be useful also elsewhere, all the more so because our approach holds for general multimatrix models. The toolkit is a noncommutative calculus on the free algebra that allows to describe the generator of the renormalization group flow -- a noncommutative Laplacian introduced here -- in terms of Voiculescu's cyclic gradient and Rota-Sagan-Stein noncommutative derivative. We explore the algebraic structure of the Functional Renormalization Group Equation and, as an application of this formalism, we find the $β$-functions, identify the fixed points in the large-$N$ limit and obtain the critical exponents of $2$-dimensional geometries in two different signatures.

math-ph↗

arXiv2020

Computing the spectral action for fuzzy geometries: from random noncommutative geometry to bi-tracial multimatrix models

A fuzzy geometry is a certain type of spectral triple whose Dirac operator crucially turns out to be a finite matrix. This notion was introduced in [J. Barrett, J. Math. Phys. 56, 082301 (2015)] and accommodates familiar fuzzy spaces like spheres and tori. In the framework of random noncommutative geometry, we use Barrett's characterization of Dirac operators of fuzzy geometries in order to systematically compute the spectral action $S(D)= \mathrm{Tr} f(D)$ for $2n$-dimensional fuzzy geometries. In contrast to the original Chamseddine-Connes spectral action, we take a polynomial $f$ with $f(x)\to \infty$ as $ |x|\to\infty$ in order to obtain a well-defined path integral that can be stated as a random matrix model with action of the type $S(D)=N \cdot \mathrm{tr}\, F+\textstyle\sum_i \mathrm{tr}\,A_i \cdot \mathrm{tr} \,B_i $, being $F,A_i $ and $B_i $ noncommutative polynomials in $2^{2n-1}$ complex $N\times N$ matrices that parametrize the Dirac operator $D$. For arbitrary signature---thus for any admissible KO-dimension---formulas for 2-dimensional fuzzy geometries are given up to a sextic polynomial, and up to a quartic polynomial for 4-dimensional ones, with focus on the octo-matrix models for Lorentzian and Riemannian signatures. The noncommutative polynomials $F,A_i $ and $B_i$ are obtained via chord diagrams and satisfy: independence of $N$; self-adjointness of the main polynomial $F$ (modulo cyclic reordering of each monomial); also up to cyclicity, either self-adjointness or anti-self-adjointness of $A_i $ and $B_i $ simultaneously, for fixed $i$. Collectively, this favors a free probabilistic perspective for the large-$N$ limit we elaborate on.

math-ph↗

arXiv2019

Graph calculus and the disconnected-boundary Schwinger-Dyson equations of quartic tensor field theories

Tensor field theory (TFT) focuses on quantum field theory aspects of random tensor models, a quantum-gravity-motivated generalisation of random matrix models. The TFT correlation functions have been shown to be classified by graphs that describe the geometry of the boundary states, the so-called boundary graphs. These graphs can be disconnected, although the correlation functions are themselves connected. In a recent work, the Schwinger-Dyson equations for an arbitrary albeit connected boundary were obtained. Here, we introduce the multivariable graph calculus in order to derive the missing equations for all correlation functions with disconnected boundary, thus completing the Schwinger-Dyson pyramid for quartic melonic (`pillow'-vertices) models in arbitrary rank. We first study finite group actions that are parametrised by graphs and build the graph calculus on a suitable quotient of the monoid algebra $A[G]$ corresponding to a certain function space $A$ and to the free monoid $G$ in finitely many graph variables; a derivative of an element of $A[G]$ with respect to a graph yields its corresponding group action on $A$. The present result and the graph calculus have three potential applications: the non-perturbative large-$N$ limit of tensor field theories, the solvability of the theory by using methods that generalise the topological recursion to the TFT setting and the study of `higher dimensional maps' via Tutte-like equations. In fact, we also offer a term-by-term comparison between Tutte equations and the present Schwinger-Dyson equations.

math-ph↗

arXiv2018

The full Schwinger-Dyson tower for random tensor models

We treat random rank-$D$ tensor models as $D$-dimensional quantum field theories---tensor field theories (TFT)---and review some of their non-perturbative methods. We classify the correlation functions of complex tensor field theories by boundary graphs, sketch the derivation of the Ward-Takahashi identity and stress its relevance in the derivation of the tower of exact, analytic Schwinger-Dyson equations for all the correlation functions (with connected boundary) of TFTs with quartic pillow-like interactions.

hep-th↗