Universal Charge Diffusion and the Butterfly Effect in Holographic Theories

We study charge diffusion in holographic scaling theories with a particle-hole symmetry. We show that these theories have a universal regime in which the diffusion constant is given by $D_c=C{v}_B^2/(2\pi T)$, where $v_B$ is the velocity of the butterfly effect. The constant of proportionality $C$ depends only on the scaling exponents of the infrared theory. Our results suggest an unexpected connection between transport at strong coupling and quantum chaos.