  Membrane paradigm is a powerful tool to study properties of black hole
horizons. We first explore the properties of the nonlinear electromagnetic
membrane of black holes. For a general nonlinear electrodynamics field, we show
that the conductivities of the horizon usually have off-diagonal components and
depend on the normal electric and magnetic fields on the horizon. Via the
holographic duality, we find a model-independent expression for the holographic
DC conductivities of the conserved current dual to a probe nonlinear
electrodynamics field in a neutral and static black brane background. It shows
that these DC conductivities only depend on the geometric and electromagnetic
quantities evaluated at the horizon. We can also express the DC conductivities
in terms of the temperature, charge density and magnetic field in the boundary
theory, as well as the values of the couplings in the nonlinear electrodynamics
at the horizon.
