arXiv subjects

Razvan Gurau

Publications and source records attributed to Razvan Gurau.

At least 19 recordsLinked to original sources

arXiv2026

From path integral quantization to stochastic quantization: a pedestrian's journey

We give two novel proofs that the path integral and stochastic quantizations of generic scalar Euclidean quantum field theories are equivalent. Our proofs rely on Taylor interpolations indexed by forests, in the fashion of constructive field theory. The first proof works at the level of individual terms in the Feynman expansion, with the forests appearing as spanning forests in Feynman graphs. The second one works at the level of the path integral and avoids the full expansion of the Feynman perturbation series.

math-ph↗

arXiv2026

Towards a quantitative characterization of gravitational universality classes for order-4 random tensor models

Random tensor models can be used as combinatorial devices to generate Euclidean dynamical triangulations. A physical continuum limit of dynamical triangulations requires a suitable generalization of the double-scaling limit of random matrices. This limit corresponds to a fixed point of a pregeometric Renormalization Group flow in which the tensor size $N$ serves as the Renormalization Group scale. We search for corresponding fixed points in order-4 random tensor models associated to dynamical triangulations in 4 dimensions. In a $O(N)^{\otimes 4}$ symmetric setting, we discuss the resulting phase portrait as a function of the regulator parameters. We optimize our results, identifying parameter values for which the results are minimally sensitive to parameter changes. We find three fixed-point candidates: only one of them is real across the entire parameter range, but only has two relevant directions. This should be contrasted with the university class of the Reuter fixed point in continuum quantum gravity, very likely characterized by three relevant directions. We conclude that simple combinatorial models of Euclidean triangulations and the Reuter fixed point most likely lie in different universality classes.

gr-qc↗

arXiv2025

The large $N$ factorization does not hold for arbitrary multi-trace observables in random tensors

We consider real tensors of order $D$, that is $D$-dimensional arrays of real numbers $T_{a^1a^2 \dots a^D}$, where each index $a^c$ can take $N$ values. The tensor entries $T_{a^1a^2 \dots a^D}$ have no symmetry properties under permutations of the indices. The invariant polynomials built out of the tensor entries are called trace invariants. We prove that for a Gaussian random tensor with $D\ge 3$ indices (that is such that the entries $T_{a^1a^2 \dots a^D}$ are independent identically distributed Gaussian random variables) the cumulant, or connected expectation, of a product of trace invariants is not always suppressed in scaling in $N$ with respect to the product of the expectations of the individual invariants. Said otherwise, not all the multi-trace expectations factor at large $N$ in terms of the single-trace ones and the Gaussian scaling is not subadditive on the connected components. This is in stark contrast to the $D=2$ case of random matrices in which the multi-trace expectations always factor at large $N$. The best one can do for $D\ge 3$ is to identify restricted families of invariants for which the large $N$ factorization holds and we check that this indeed happens when restricting to the family of melonic observables, the dominant family in the large $N$ limit.

math-ph↗

arXiv2025

Asymptotic safety meets tensor field theory: towards a new class of gravity-matter systems

Combining asymptotically safe quantum gravity with a tensor field theory, we exhibit the first example of a theory with gravity and scalar fields in four dimensions which may realize asymptotic safety at a non-vanishing value of the scalar quartic coupling. We first present (further) evidence that in the asymptotic-safety paradigm, quantum fluctuations of gravity generically screen the quartic couplings in (multi-)scalar models. For a tensor field theory in which the scalar field transforms under an internal $O(N)^3$ symmetry, this has the effect of replacing asymptotic freedom, recently discovered at large $N$ on a fixed flat background, by an interacting fixed point in the presence of quantum gravity. The fixed point originates from the competition between the effects of the matter self-interactions which, contrary to the usual scalar models, are antiscreening, and the screening gravitational effects.

hep-th↗

arXiv2024

Addendum: Long-range multi-scalar models at three loops

We correct the computation of one Feynman diagram in the three-loop beta functions for the long-range quartic multi-scalar model, originally presented in (2020 J. Phys. A: Math. Theor. 53 445008) [arXiv:2007.04603]. The correction requires the use of a different method than in the original paper, and we give here full details about the method. We then report the updated numerics for critical exponents of the Ising model, vector model, cubic model and bifundamental model. Mathematica files for the numerical evaluation of the corrected diagram are provided in ancillary.

hep-th↗

arXiv2024

Free cumulants and freeness for unitarily invariant random tensors

We address the question of the asymptotic description of random tensors that are local-unitary invariant, that is, invariant by conjugation by tensor products of independent unitary matrices. We consider both the mixed case of a tensor with $D$ inputs and $D$ outputs, and the case where there is a factorization between the inputs and outputs, called pure, which includes the random tensor models extensively studied in the physics literature. The finite size and asymptotic moments are defined using correlations of certain invariant polynomials encoded by $D$-tuples of permutations, up to relabeling equivalence. Finite size free cumulants associated to the expectations of these invariants are defined through invertible finite size moment-cumulants formulas. Two important cases are considered asymptotically: pure random tensors that scale like a complex Gaussian, and mixed random tensors that scale like a Wishart tensor. In both cases, we derive a notion of tensorial free cumulants associated to first order invariants, through moment-cumulant formulas involving summations over non-crossing permutations. The pure and mixed cases involve the same combinatorics, but differ by the invariants that define the distribution at first order. In both cases, the tensorial free-cumulants of a sum of two independent tensors are shown to be additive. A preliminary discussion of higher orders is provided. Tensor freeness is then defined as the vanishing of mixed first order tensorial free cumulants. The equivalent formulation at the level of asymptotic moments is derived in the pure and mixed cases, and we provide an algebraic construction of tensorial probability spaces, which generalize non-commutative probability spaces: random tensors converge in distribution to elements of these spaces, and tensor freeness of random variables corresponds to tensor freeness of the subspaces they generate.

math-ph↗

arXiv2024

Coupling renormalization flow in the strongly interacting regime of an asymptotically free quantum field theory in four dimensions

We consider a scalar quantum field theory with global $O(N)^3$ symmetry in four Euclidean dimensions and solve it numerically in closed form in the large-N limit. For imaginary tetrahedral coupling the theory is asymptotically free, with stable and real quantum effective action. We demonstrate the dynamical build-up of a strong interaction as the correlation length increases in a regime where the coupling renormalization flow remains well defined in the infrared. This is in contrast to perturbative results of asymptotically free theories, which predict that the coupling becomes ill-defined in the infrared, like in quantum chromodynamics. These properties make the model an important laboratory for the study of strong-coupling phenomena in quantum field theory from first principles.

hep-th↗

arXiv2024

Dynamic critical exponent in quantum long-range models

Quantum long-range models at zero temperature can be described by fractional Lifshitz field theories, that is, anisotropic models whose actions are short-range in time and long-range in space. In this paper we study the renormalization of fractional Lifshitz field theories with weakly relevant cubic or quartic self-interactions. Their nontrivial infrared fixed points exhibit Lifshitz scale invariance, and we compute the lowest-order corrections to the dynamic critical exponent.

hep-th↗

arXiv2024

Quantum Gravity and Random Tensors

Random tensors are the natural generalization of random matrices to higher order objects. They provide generating functions for random geometries and, assuming some familiarity with random matrix theory and quantum field theory, we discuss in the first part of this note the applications of such models to quantum gravity. In a second part we review tensor field theories, that is standard field theories in $\mathbb{R}^d$ but with tensor fields, which lead to a new family of large $N$ conformal field theories relevant for the study of the $AdS/CFT$ correspondence.

hep-th↗

arXiv2024

Finite-size versus finite-temperature effects in the critical long-range $O(N)$ model

In this paper we consider classical and quantum versions of the critical long-range $O(N)$ model, for which we study finite-size and finite-temperature effects, respectively, at large $N$. First, we consider the classical (isotropic) model, which is conformally invariant at criticality, and we introduce one compact spatial direction. We show that the finite size dynamically induces an effective mass and we compute the one-point functions for bilinear primary operators with arbitrary spin and twist. Second, we study the quantum model, mapped to a Euclidean anisotropic field theory, local in Euclidean time and long-range in space, which we dub \emph{fractional Lifshitz field theory}. We show that this model admits a fixed point at zero temperature, where it displays anisotropic Lifshitz scaling, and show that at finite temperature a thermal mass is induced. We then compute the one-point functions for an infinite family of bilinear scaling operators. In both the classical and quantum model, we find that, as previously noted for the short-range $O(N)$ model in [arXiv:1802.10266], the large-$N$ two-point function contains information about the one-point functions, not only of the bilinear operators, but also of operators that appear in the operator product expansion of two fundamental fields only at subleading order in $1/N$, namely powers of the Hubbard-Stratonovich intermediate field.

hep-th↗

arXiv2023

Asymptotic freedom in a strongly interacting scalar quantum field theory in four Euclidean dimensions

We show that scalar quantum field theory in four Euclidean dimensions with global $O(N)^3$ symmetry and imaginary tetrahedral coupling is asymptotically free and bounded from below in the large-N limit. While the Hamiltonian is non-Hermitian, the full quantum effective action for the large-N theory only depends on the square of that coupling which is real. A perturbative analysis uncovers that the renormalization group flow of the quartic couplings connects a Gaussian ultraviolet fixed point to a strongly interacting theory in the infrared. This realizes a renormalizable field theory which exhibits non-trivial dynamics, such as direct scattering, while still being analytically tractable also non-perturbatively. Our findings open up a way to address outstanding problems in strongly coupled theories from first principles.

hep-th↗

arXiv2022

The small-$N$ series in the zero-dimensional $O(N)$ model: constructive expansions and transseries

We consider the 0-dimensional quartic $O(N)$ vector model and present a complete study of the partition function $Z(g,N)$ and its logarithm, the free energy $W(g,N)$, seen as functions of the coupling $g$ on a Riemann surface. Using constructive field theory techniques we prove that both $Z(g,N)$ and $W(g,N)$ are Borel summable functions along all the rays in the cut complex plane $\mathbb{C}_{\pi} =\mathbb{C}\setminus \mathbb{R}_-$. We recover the transseries expansion of $Z(g,N)$ using the intermediate field representation. We furthermore study the small-$N$ expansions of $Z(g,N)$ and $ W(g,N)$. For any $g=|g| e^{\imath \varphi}$ on the sector of the Riemann surface with $|\varphi|<3\pi/2$, the small-$N$ expansion of $Z(g,N)$ has infinite radius of convergence in $N$ while the expansion of $W(g,N)$ has a finite radius of convergence in $N$ for $g$ in a subdomain of the same sector. The Taylor coefficients of these expansions, $Z_n(g)$ and $W_n(g)$, exhibit analytic properties similar to $Z(g,N)$ and $W(g,N)$ and have transseries expansions. The transseries expansion of $Z_n(g)$ is readily accessible: much like $Z(g,N)$, for any $n$, $Z_n(g)$ has a zero- and a one-instanton contribution. The transseries of $W_n(g)$ is obtained using M\"oebius inversion and summing these transseries yields the transseries expansion of $W(g,N)$. The transseries of $W_n(g)$ and $W(g,N)$ are markedly different: while $W(g,N)$ displays contributions from arbitrarily many multi-instantons, $W_n(g)$ exhibits contributions of only up to $n$-instanton sectors.

hep-th↗

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

Duality of Orthogonal and Symplectic Random Tensor Models

The groups $O(N)$ and $Sp(N)$ are related by an analytic continuation to negative values of $N$, $O(-N)\simeq Sp(N)$. This duality has been studied for vector models, $SO(N)$ and $Sp(N)$ gauge theories, as well as some random matrix ensembles. We extend this duality to real random tensor models of arbitrary order $D$ with no symmetry under permutation of the indices and with quartic interactions. The $N$ to $-N$ duality is shown to hold graph by graph to all orders in perturbation theory for the partition function, the free energy and the connected two point function.

math-ph↗

arXiv2022

The F-theorem in the melonic limit

The $F$-theorem states that in three dimensions the sphere free energy of a field theory must decrease between ultraviolet and infrared fixed points of the renormalization group flow, and it has been proven for unitary conformal field theories (CFTs). We consider here the long-range bosonic $O(N)^3$ model on a spherical background, at next-to-next-to-leading order of the $1/N$ expansion. The model displays four large-$N$ fixed points and we test and confirm the $F$-theorem holds in this case. This is non-trivial as one of the couplings is imaginary, and therefore the model is non-unitary at finite $N$. Despite this, several tests indicating that the large-$N$ CFTs are in fact unitary have been performed: for instance all the OPE coefficients computed so far in the large-$N$ limit are real, and the spectrum of bilinear operators is real and above unitarity bounds. Our result, namely that the F theorem holds at large $N$, can be viewed as further indication that such theories are unitary. As an added bonus, we show how conformal partial waves expansions in conformal field theory can be used to resum infinite classes of vacuum diagrams. Non-perturbatively, the jump in the value of the free energy has the interpretation of the inclusion at the ultraviolet fixed point of an extra non-normalizable contribution in the conformal partial wave expansion. This can be seen in perturbation theory as the reversal of the sign of an infinite class of diagrams due to the flow of a coupling constant.

hep-th↗

arXiv2022

The tensor Harish-Chandra--Itzykson--Zuber integral II: detecting entanglement in large quantum systems

We consider the recently introduced generalization of the Harish-Chandra--Itzykson--Zuber integral to tensors and discuss its asymptotic behavior when the characteristic size N of the tensors is taken to be large. This study requires us to make assumptions on the scaling with N of the external tensors. We analyze a two-parameter class of asymptotic scaling ansätze, uncovering several non-trivial asymptotic regimes. This study is relevant for analyzing the entanglement properties of multipartite quantum systems. We discuss potential applications of our results to this domain, in particular in the context of randomized local measurements.

math-ph↗

arXiv2022

The tensor Harish-Chandra-Itzykson-Zuber integral I: Weingarten calculus and a generalization of monotone Hurwitz numbers

We study a generalization of the Harish-Chandra - Itzykson - Zuber integral to tensors and its expansion over trace-invariants of the two external tensors. This gives rise to natural generalizations of monotone double Hurwitz numbers, which count certain families of constellations. We find an expression of these numbers in terms of monotone simple Hurwitz numbers, thereby also providing expressions for monotone double Hurwitz numbers of arbitrary genus in terms of the single ones. We give an interpretation of the different combinatorial quantities at play in terms of enumeration of nodal surfaces. In particular, our generalization of Hurwitz numbers is shown to enumerate certain isomorphism classes of branched coverings of a bouquet of $D$ 2-spheres that touch at one common non-branch node.

math.CO↗

arXiv2020

The tri-fundamental quartic model

We consider a multi-scalar field theory with either short-range or long-range free action and with quartic interactions that are invariant under $O(N_1)\times O(N_2) \times O(N_3)$ transformations, of which the scalar fields form a tri-fundamental representation. We study the renormalization group fixed points at two loops at finite $N$ and in various large-$N$ scaling limits for small $ε$, the latter being either the deviation from the critical dimension or from the critical scaling of the free propagator. In particular, for the homogeneous case $N_i = N$ for $i=1,2,3$, we study the subleading corrections to previously known fixed points. In the short-range model, for $εN^2\gg 1$, we find complex fixed points with non-zero tetrahedral coupling, that at leading order reproduce the results of arXiv:1707.03866 ; the main novelty at next-to-leading order is that the critical exponents acquire a real part, thus allowing a correct identification of some fixed points as IR stable. In the long-range model, for $εN \ll 1 $, we find again complex fixed points with non-zero tetrahedral coupling, that at leading order reproduce the line of stable fixed points of arXiv:1903.03578; at next-to-leading order, this is reduced to a discrete set of stable fixed points. One difference between the short-range and long-range cases is that, in the former the critical exponents are purely imaginary at leading-order and gain a real part at next-to-leading order, while for the latter the situation is reversed.

hep-th↗