This talk addresses the low energy physics of the Sachdev-Ye-Kitaev model, a paradigm of strongly interacting (Majorana) quantum matter. A salient feature of this system is its exceptionally high degree of symmetry under reparameterizations of physical time. At low energies this symmetry is spontaneously broken and the ensuing infinite dimensional Goldstone mode manifold takes strong influence on all physical observables. We will discuss the effects of these fluctuations on the example of the so-called out of time ordered correlation functions, diagnostic tools to describe both manifestations of quantum chaos in the system and its conjectured duality to an AdS2 gravitational bulk. While previous work predicts exponential decay of these correlations in time our main finding is that at large time scales non-perturbative Goldstone mode fluctuations generate a crossover to power law behavior. This phenomenon must have ramifications in the physics of the holographic bulk which, however, we do not understand at present.