Results of experimental and theoretical studies of the arc discharge which slides over the surface of a conductor are reported. Experiments were performed in air and argon ambients at various pressures. It is found that the velocity of the discharge plasma front depends linearly on the strength of the average electric field ahead of the front. A physical model of this phenomenon for the range of electric fields near the discharge threshold was developed. It is demonstrated that the mechanism related to thermal conductivity is responsible for the plasma front propagation.