Melonic Turbulence

TL;DR

This paper introduces melonic turbulence, a novel phenomenon in non-linear random flows, demonstrating how initial excitations spread over many modes through a tensor-theoretic approach.

Contribution

It applies random tensor theory to non-linear resonant systems, revealing a leading regime governed by melonic graphs and demonstrating turbulent cascade behavior.

Findings

Initial excitation spreads over more modes during finite time.

Melonic graphs dominate the perturbation theory regime.

Turbulent cascade observed in the melonic approximation.

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 Schr\"odinger 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.