New Limits for Large $N$ Matrix and Tensor Models: Large $D$, Melons and Applications

Abstract

Large $N$ matrix models play an important role in modern theoretical physics, ranging from quantum chromodynamics to string theory and holography. However, they remain a difficult technical challenge because in most cases it is not known how to perform the sum over planar graphs, which dominate the models at large $N$. In this thesis, we study large $D$ matrix models, which provide a framework to build new limits for matrix models in which the sum over planar graphs simplifies when $D$ is large. The basic degrees of freedom are real matrices of size $N\times N$ with $r$ additional indices of range $D$. They can be interpreted as a real tensor of rank $R=r+2$ with indices of different ranges, making a compelling connection with tensor models. We define a new large $D$ limit for the sum over Feynman graphs of fixed genus in matrix models, based on an enhanced large $D$ scaling of the coupling constants. Then, we show that the resulting large $D$ expansion is well-defined and organized according to a half-integer called the index. When $N=D$, the result provides a new large $N$ limit for general $\text{O}(N{)}^R$ invariant tensor models. In the large $D$ limit, the sum over planar graphs of large $N$ matrix models simplifies to a non-trivial sum over generalized melonic graphs. This class of graphs extends the one obtained in tensor models with standard scaling and allows for a wider class of interactions, including all the maximally single-trace terms. The general classification of generalized melonic graphs remains an open problem. However, in the case of the complete interaction of order $R+1$ for $R$ a prime number, we identify them in detail and demonstrate that they exhibit the same important features as the SYK model with $q=(R+1)$-fold random interactions, including the emergent conformal symmetry in the infrared regime and maximal chaos.

arXiv commentPedagogical part of Ph.D. thesis

References in corpus (103)

• Large N Field Theories, String Theory and Gravity
• The World as a Hologram
• M Theory As A Matrix Model: A Conjecture
• A bound on chaos
• Gapless Spin-Fluid Ground State in a Random Quantum Heisenberg Magnet
• Comments on the Sachdev-Ye-Kitaev model
• Black holes and the butterfly effect
• AdS Dual of the Critical O(N) Vector Model
• Quasinormal Modes of AdS Black Holes and the Approach to Thermal Equilibrium
• The Spectrum in the Sachdev-Ye-Kitaev Model
• Localized shocks
• Bekenstein-Hawking Entropy and Strange Metals
• Multiple Shocks
• Two-dimensional conformal field theory and the butterfly effect
• Spin-Glass Theory for Pedestrians
• Thermoelectric transport in disordered metals without quasiparticles: the SYK models and holography
• Holographic metals and the fractionalized Fermi liquid
• Critical behavior of colored tensor models in the large N limit
• Uniqueness and universality of the Brownian map
• Uncolored Random Tensors, Melon Diagrams, and the SYK Models
• Colored Tensor Models - a Review
• A Generalization of Sachdev-Ye-Kitaev
• Non-Fermi liquid regime of a doped Mott insulator
• Random tensor models in the large N limit: Uncoloring the colored tensor models
• AdS${}_2$ holography and the SYK model
• The large D limit of General Relativity
• Mean-Field Theory of a Quantum Heisenberg Spin Glass
• The complete $1/N$ expansion of a SYK--like tensor model
• Holographic Thermalization, stability of AdS, and the Fermi-Pasta-Ulam-Tsingou paradox
• Dynamical Aspects of Large N Reduced Models
• Melons are branched polymers
• Double Scaling in Tensor Models with a Quartic Interaction
• Models of AdS_2 Backreaction and Holography
• Renormalization group, secular term resummation and AdS (in)stability
• A note on Gromov-Hausdorff-Prokhorov distance between (locally) compact measure spaces
• Towards a 2d QFT Analog of the SYK Model
• Diagrammatics of a colored SYK model and of an SYK-like tensor model, leading and next-to-leading orders
• A generalization of the Virasoro algebra to arbitrary dimensions
• Linearized Group Field Theory and Power Counting Theorems
• Bosonic Tensor Models at Large $N$ and Small $\epsilon$
• The Tensor Track, III
• Tensorial methods and renormalization in Group Field Theories
• The 1/N Expansion of Tensor Models Beyond Perturbation Theory
• Notes on Melonic $O(N{)}^{q-1}$ Tensor Models
• Phase Diagram of Planar Matrix Quantum Mechanics, Tensor, and Sachdev-Ye-Kitaev Models
• Resonant dynamics and the instability of anti-de Sitter spacetime
• Revisiting random tensor models at large N via the Schwinger-Dyson equations
• Spectra of Operators in Large $N$ Tensor Models
• Flowing in Group Field Theory Space: a Review
• Invitation to Random Tensors
• New 1/N expansions in random tensor models
• The 1/N expansion of multi-orientable random tensor models
• Universal quasinormal modes of large D black holes
• The Multi-Orientable Random Tensor Model, a Review
• Multi-matrix models and emergent geometry
• Scaling limits of random trees and planar maps
• A Proposal for Strings at D>1
• SYK Models and SYK-like Tensor Models with Global Symmetry
• Renormalization: an advanced overview
• Gauge Invariants, Correlators and Holography in Bosonic and Fermionic Tensor Models
• Towards a double-scaling limit for tensor models: probing sub-dominant orders
• Enhancing non-melonic triangulations: A tensor model mixing melonic and planar maps
• 2PI effective action for the SYK model and tensor field theories
• Matrix Model Perturbed by Higher Order Curvature Terms
• The $1/N$ expansion of the symmetric traceless and the antisymmetric tensor models in rank three
• The double scaling limit of the multi-orientable tensor model
• Phases Of Melonic Quantum Mechanics
• Tensorial Gross-Neveu models
• Exact lowest-Landau-level solutions for vortex precession in Bose-Einstein condensates
• The $1/N$ expansion of tensor models with two symmetric tensors
• Random Tensors and Quantum Gravity
• Tensor models from the viewpoint of matrix models: the case of loop models on random surfaces
• Scaling Functions for Baby Universes in Two-Dimensional Quantum Gravity
• Combinatorial study of graphs arising from the Sachdev-Ye-Kitaev model
• Non-Gaussian disorder average in the Sachdev-Ye-Kitaev model
• SYK-like tensor quantum mechanics with $\mathrm{Sp}(N)$ symmetry
• The Tensor Theory Space
• Constructive Tensor Field Theory
• Large $N$ Limits in Tensor Models: Towards More Universality Classes of Colored Triangulations in Dimension $d\ge 2$
• Multi-critical behaviour of 4-dimensional tensor models up to order 6
• Gauge Theories, D-Branes and Holography
• Exact Renormalisation Group Equations and Loop Equations for Tensor Models
• Two infinite families of resonant solutions for the Gross-Pitaevskii equation
• D-Brane Probes in the Matrix Model
• Majorana Fermion Quantum Mechanics for Higher Rank Tensors
• Emergent Space and the Example of AdS_5XS^5
• Solvable cubic resonant systems
• Stringy effects in scrambling
• Sketches of emergent geometry in the gauge/gravity duality
• Examples of Emergent Type IIB Backgrounds from Matrices
• Bi-Local Holography in the SYK Model: Perturbations
• Conformality of $1/N$ corrections in SYK-like models
• On Matrix Geometry and Effective Actions
• Emergent D4-Brane Background from D-Particles
• Renormalizable Models in Rank $d\ge 2$ Tensorial Group Field Theory
• Emergent D5-brane Background from D-strings
• Weakly turbulent solutions for the cubic defocusing nonlinear Schrödinger equation
• Vector models in the large $N$ limit: a few applications
• A bijection for rooted maps on orientable surfaces
• On the continuous resonant equation for NLS: I. Deterministic analysis
• Black hole entropy and quantum mechanics
• Tensor models with generalized melonic interactions
• Tensor models from the viewpoint of matrix models: the case of the Gaussian distribution

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