Consider a wireless network of transmitter-receiver pairs. The transmitters adjust their powers to maintain a particular SINR target in the presence of interference from neighboring transmitters. In this paper we analyze the optimal power vector that may achieve this target in the presence of randomness in the network. In particular, we address the scenario where the receiver pairs may be located in one of two distinct distances from their serving transmitter base. We apply concepts from random matrix theory to evaluate the asymptotic mean optimal power per link. Our analytical results show substantial agreement with numerically generated networks, not only in one-dimensional network arrays but also in two dimensional network geometries. Notably, we observe that the optimal power in random networks does not go to infinity in a continuous fashion as in regular grids. Rather, beyond a certain point, no finite power solution exists.