Seminar: "Melonic turbulence" by Mr. Guillaume Valette

Speaker: Mr. Valette is currently in his last year as a Ph.D. student at Univ. Libre de Bruxelles, Belgium.

Abstract. I will speak about a new application of random tensor theory to the study of non-linear random flows in many variables (based on arXiv:1810.01848). The class of systems that will be considered are described by a non-linear and resonant system of equations. I will illustrate how such systems can emerge as weakly non-linear approximations to problems whose linearized perturbations have a highly resonant spectrum of frequencies. A key question for solutions of resonant systems is the understanding of energy cascades and turbulence. In our work, we studied these solutions in perturbative theory. Then, we performed a Gaussian averaging over ensembles of resonant systems, that is, over the tensor coupling and the initial conditions, which gives rise to Feynman-like graphs. In the limit of many initially excited modes, we proved that the dominant graphs at each order in perturbation theory are the melonic graphs, which generically dominate in random tensor theory. Finally, we showed that when we restrict the non-linear flow to the corresponding melonic approximation, the initial excitation spreads over more modes, at least within a certain initial time interval, as expected in a turbulent cascade. This phenomenon was called melonic turbulence.