Melonic Turbulence

Abstract

We propose a new application of random tensor theory to studies of non-linear random flows in many variables. Our focus is on non-linear resonant systems which often emerge as weakly non-linear approximations to problems whose linearized perturbations possess highly resonant spectra of frequencies (non-linear Schrödinger equations for Bose-Einstein condensates in harmonic traps, dynamics in Anti-de Sitter spacetimes, etc). We perform Gaussian averaging both for the tensor coupling between modes and for the initial conditions. In the limit when the initial configuration has many modes excited, we prove that there is a leading regime of perturbation theory governed by the melonic graphs of random tensor theory. Restricting the flow equation to the corresponding melonic approximation, we show that at least during a finite time interval, the initial excitation spreads over more modes, as expected in a turbulent cascade. We call this phenomenon melonic turbulence.

arXiv comment54 pages, 20 figures

References in corpus (49)

• Comments on the Sachdev-Ye-Kitaev model
• 2D Gravity and Random Matrices
• The Spectrum in the Sachdev-Ye-Kitaev Model
• On weakly turbulent instability of anti-de Sitter space
• Critical behavior of colored tensor models in the large N limit
• Uncolored Random Tensors, Melon Diagrams, and the SYK Models
• Colored Tensor Models - a Review
• Mode-Coupling Approximations, Glass Theory and Disordered Systems
• The complete 1/N expansion of colored tensor models in arbitrary dimension
• Colored Group Field Theory
• The 1/N expansion of colored tensor models in arbitrary dimension
• The 1/N expansion of colored tensor models
• Top eigenvalue of a random matrix: large deviations and third order phase transition
• The complete $1/N$ expansion of a SYK--like tensor model
• Holographic Thermalization, stability of AdS, and the Fermi-Pasta-Ulam-Tsingou paradox
• Double Scaling in Tensor Models with a Quartic Interaction
• Renormalization group, secular term resummation and AdS (in)stability
• $O(N)$ Random Tensor Models
• Renormalization of an SU(2) Tensorial Group Field Theory in Three Dimensions
• Renormalization, averaging, conservation laws and AdS (in)stability
• The 1/N Expansion of Tensor Models Beyond Perturbation Theory
• Phase Diagram of Planar Matrix Quantum Mechanics, Tensor, and Sachdev-Ye-Kitaev Models
• Resonant dynamics and the instability of anti-de Sitter spacetime
• Renormalization of Tensorial Group Field Theories: Abelian U(1) Models in Four Dimensions
• Flowing in Group Field Theory Space: a Review
• The 1/N expansion of multi-orientable random tensor models
• Multi-orientable Group Field Theory
• The $1/N$ expansion of the symmetric traceless and the antisymmetric tensor models in rank three
• A Global View on The Search for de-Sitter Vacua in (type IIA) String Theory
• Exact lowest-Landau-level solutions for vortex precession in Bose-Einstein condensates
• SYK-like tensor quantum mechanics with $\mathrm{Sp}(N)$ symmetry
• On the Cubic Lowest Landau Level Equation
• Constructive Tensor Field Theory
• Delayed collapses of BECs in relation to AdS gravity
• Multi-critical behaviour of 4-dimensional tensor models up to order 6
• Conformal flow on $S^3$ and weak field integrability in AdS${}_4$
• More on the New Large $D$ Limit of Matrix Models
• Large $N$ limit of irreducible tensor models: $O(N)$ rank-$3$ tensors with mixed permutation symmetry
• Exact Renormalisation Group Equations and Loop Equations for Tensor Models
• Two infinite families of resonant solutions for the Gross-Pitaevskii equation
• Solvable cubic resonant systems
• Maximally rotating waves in AdS and on spheres
• The effective equation method
• Weakly turbulent solutions for the cubic defocusing nonlinear Schrödinger equation
• An explicit formula for the cubic Szegö equation
• The Szegö Cubic Equation
• Effective integrable dynamics for some nonlinear wave equation
• Stationary states of the cubic conformal flow on ${\mathbb{S}}^3$
• Ground state of the conformal flow on ${\mathbb{S}}^3$

Cited by in corpus (8)

• Phases Of Melonic Quantum Mechanics
• Ward identity violation for melonic $T^4$-truncation
• Resonant Hamiltonian systems and weakly nonlinear dynamics in AdS spacetimes
• Fermi-Pasta-Ulam phenomena and persistent breathers in the harmonic trap
• Melonic dominance and the largest eigenvalue of a large random tensor
• Flowing in discrete gravity models and Ward identities: A review
• Scattering of the three-dimensional cubic nonlinear Schrödinger equation with partial harmonic potentials
• Random coupling model of turbulence as a classical Sachdev-Ye-Kitaev model