Ultra-dense cloud radio access networks (C-RANs) is one of the architectures that will be critical components of the next-generation wireless systems. In a C-RAN architecture, an amorphous cellular framework, where each user connects to a few nearby remote radio heads (RRHs) to form its own cell, appears to be promising. In this paper, we study the ergodic capacity of such amorphous cellular networks at high signal-tonoise ratios (SNRs) where we model the distribution of the RRHs by a Poisson point process. We derive tractable approximations of the ergodic capacity at high-SNRs for arbitrary antenna configurations, and tight lower bounds for the ergodic capacity when the numbers of antennas are the same at both ends of the link. In contrast to prior works on distributed antenna systems, our results are derived based on random matrix theory and involve only standard functions which can be much more easier evaluated. The impact of the system parameters on the ergodic capacity is investigated. By leveraging our analytical results, we propose two efficient scheduling algorithms for RRH selection for energy-efficient transmission. Our algorithms offer a substantial improvement in energy efficiency compared to the strategy of connecting a fixed number of RRHs to each user.