Melonic Turbulence

Stéphane Dartois, Oleg Evnin, Luca Lionni, Vincent Rivasseau, Guillaume Valette

Research output: Contribution to journalArticle

1 Citation (Scopus)
10 Downloads (Pure)


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.

Original languageEnglish
Article number374
Pages (from-to)1179-1228
Number of pages54
JournalComm. Math. Phys.
Issue number2
Publication statusPublished - Mar 2020

Bibliographical note

54 pages, 20 figures


  • math-ph
  • hep-th
  • math.AP
  • math.MP


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