Outer bound to the capacity scaling of three dimensional wireless networks

It is shown that the capacity scaling of three- dimensional wireless networks is subject to a degrees of freedom limitation which is due to the propagation laws of electromagnetic waves. By distributing uniformly an order of n users wishing to establish pairwise independent communications at fixed wavelength lambda inside a domain of (normalized) volume of the order of n, there are an order of n communication requests originating from the central half of the domain to its outer half. Physics dictates that the number of independent information channels across these two regions is only of the order of n^2/3, so the peruser information capacity must follow an inverse-cube root of n law.