Quenched disorder slows down the scrambling of quantum information. Using a bottom-up approach, we formulate a kinetic theory of scrambling in a correlated metal near a superconducting transition, following the scrambling dynamics as the impurity scattering rate is increased. Within this framework, we rigorously show that the butterfly velocity $v$ is bounded by the light-cone velocity $v_{\mathrm{lc}}$ set by the Fermi velocity. We analytically identify a disorder-driven dynamical transition occurring at small but finite disorder strength between a spreading of information characterized at late times by a discontinuous shock wave propagating at the maximum velocity $v_{\mathrm{lc}}$, and a smooth traveling wave belonging to the Fisher or Kolmogorov-Petrovsky-Piskunov (FKPP) class and propagating at a slower, if not considerably slower, velocity $v$. In the diffusive regime, we establish the relation $v^2/{\lambda }_{\mathrm{FKPP}}\sim D_{\mathrm{el}}$, where ${\lambda }_{\mathrm{FKPP}}$ is the Lyapunov exponent set by the inelastic scattering rate and $D_{\mathrm{el}}$ is the elastic diffusion constant.

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• Information scrambling
• Quantum chaos
• Quantum many-body systems
• Random & disordered media
• Strongly correlated systems