  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.
