Lower estimate of potential for a class of volume circuits

In this paper, volume circuits are considered, which are the embeddings of Boolean circuits in space. For volume circuits, the lower bound on the potential is obtained. Potential is a measure of circuit activity equal to the number of gates that produce one on a given input. It is shown that for almost all partial operators with $n$ inputs and $m$ outputs, the potential of the volume circuit that implements them is not less than $ \frac{m \sqrt[3]{d}}{\min^{2/3}(m, log_2 d)} $, where $d$ is the size of the domain.