Highly nonlinear current–voltage (J∝Eα) relations with voltage-limiting characteristics are observed for Mg-doped lanthanum calcium manganite polycrystalline ceramics with nonlinearity coefficient, α = 2–9 at low-electric-field strengths of 2–5 V/mm, below magnetic transition temperatures. The current density increases with external magnetic field, so that magnetically tunable low-voltage varistors are realized. The α increases on annealing at 1375 K in atmospheres of lower pO2, and becomes more pronounced with decreasing grain size, indicating that nonlinear behavior is related to the outdiffusion of oxygen through the grain-boundary-layer regions. The increasing deficiency in Mn3+/Mn4+ pairs reduces the hole-hopping probability leading to insulating barriers in the grain-boundary-layer regions. The barrier height is lowered by the external electric field, facilitating the charge-carrier movement between the grains. Since the conduction is due to spin-polarized tunneling, the external magnetic field increases the current. © 2001 American Institute of Physics.