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Quenched disorder slows down the scrambling of quantum information. Using abottom-up approach, we formulate a kinetic theory of scrambling in a correlatedmetal near a superconducting transition, following the scrambling dynamics asthe impurity scattering rate is increased. Within this framework, we rigorouslyshow that the butterfly velocity $v$ is bounded by the light cone velocity$v_{\rm lc }$ set by the Fermi velocity. We analytically identify adisorder-driven dynamical transition occurring at small but finite disorderstrength between a spreading of information characterized at late times by adiscontinuous shock wave propagating at the maximum velocity $v_{\rm lc}$, anda smooth traveling wave belonging to the Fisher orKolmogorov-Petrovsky-Piskunov (FKPP) class and propagating at a slower, if notconsiderably slower, velocity $v$. In the diffusive regime, we establish therelation $v^2/\lambda_{\rm FKPP} \sim D_{\rm el}$ where $\lambda_{\rm FKPP}$ isthe Lyapunov exponent set by the inelastic scattering rate and $D_{\rm el}$ isthe elastic diffusion constant.

Kinetics of information scrambling in correlated electrons: disorder-driven transition from shock-wave to FKPP dynamics