We present a perturbation method to compute the out-of-time-ordered correlator in the strongly disordered Heisenberg XXZ model in the deep many-body localized regime. We characterize the discrete structure of the information propagation across the eigenstates, revealing a highly structured light cone confined by the strictly logarithmic upper and lower bounds representing the slowest and fastest scrambling available in this system. We explain these bounds by deriving the closed-form expression of the effective interaction for the slowest scrambling and by constructing the effective model of a half length for the fastest scrambling. We extend our lowest-order perturbation formulations to the higher dimensions, proposing that the logarithmic upper and lower light cones may persist in a finite two-dimensional system in the limit of strong disorder and weak hopping.

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• Information scrambling
• Many-body localization
• 2-dimensional systems
• Heisenberg model
• Perturbation theory