Fast Scramblers, Democratic Walks and Information Fields

We study a family of weighted random walks on complete graphs. These `democratic walks' turn out to be explicitly solvable, and we find the hierarchy window for which the characteristic time scale saturates the so-called fast scrambling conjecture. We show that these democratic walks describe well the properties of information spreading in systems in which every degree of freedom interacts with every other degree of freedom, such as Matrix or infinite range models. The argument is based on the analysis of suitably defined `Information fields' ($\mathcal{I}$), which are shown to evolve stochastically towards stationarity due to unitarity of the microscopic model. The model implies that in democratic systems, stabilization of one subsystem is equivalent to global scrambling. We use these results to study scrambling of infalling perturbations in black hole backgrounds, and argue that the near horizon running coupling constants are connected to entanglement evolution of single particle perturbations in democratic systems.