  The density of states reproducing the Bekenstein-Hawking entropy-area scaling
can be modeled via a nonlocal field theory. We define a diffusion process based
on the kinematics of this theory and find a spectral dimension whose flow
exhibits surprising properties. While it asymptotes four from above in the
infrared, in the ultraviolet the spectral dimension diverges at a finite
(Planckian) value of the diffusion length, signaling a breakdown of the notion
of diffusion on a continuum spacetime below that scale. We comment on the
implications of this minimal diffusion scale for the entropy bound in a
holographic and field-theoretic context.
