A nontrivial two‐dimensional stochastic model for dielectric breakdown within a parallel plate capacitor is presented for the first time. The model has been used to determine geometric properties of parallel plate discharges. Comparisons are made between these properties and known fractal properties of electrostatic discharges within cylindrical geometries. As the spacing between the plates of a capacitor increases, the value of the fractal dimension of the associated discharge structure increases from the minimum value of unity and approaches the limiting value corresponding to the case of infinite spacing. For any given spacing, this fractal exponent is equal to the exponent of first passage time for the discharge pattern to reach a given height. A study of various power law relationships governing the breakdown may provide insight into the breakdown mechanism and electrical insulating quality of various materials. The model is applicable to the breakdown of thin insulating layers of metal‐oxide‐semiconductor devices.