Decoherence and Microscopic Diffusion at SYK

Sachdev-Ye-Kitaev (SYK) or embedded random ensembles are models of $N$ fermions with random k-body interactions. They play an important role in understanding black hole dynamics, quantum chaos, and thermalization. We study out of equilibrium scenarios in these systems and show they display perfect decoherence at all times. This peculiar feature makes them very attractive in the context of the quantum-to-classical transition and the emergence of classical general relativity. Based on this feature and unitarity, we propose a rate/continuity equation for the dynamics of the $\mathcal{O}(e^N)$ microstates probabilities. The effective permutation symmetry of the models drastically reduces the number of variables, allowing for compact expressions of n-point correlation functions and entropy of the microscopic distribution. Further assuming a generalized Fermi golden rule allows finding analytic formulas for the kernel spectrum at finite $N$, providing a series of short and long time scales controlling the out of equilibrium dynamics of this model. This approach to chaos, long time scales, and $1/N$ corrections might be tested in future experiments.