Fast scramblers and ultrametric black hole horizons

We propose that fast scrambling on finite-entropy stretched horizons can be modeled by a diffusion process on an effective ultrametric geometry. A scrambling time scaling logarithmically with the entropy is obtained when the elementary transition rates saturate causality bounds on the stretched horizon. The so-defined ultrametric diffusion becomes unstable in the infinite-entropy limit. A formally regularized version can be shown to follow a particular case of the Kohlrausch law.