Section: New Results

Power control in random wireless networks

Ever since the early development stages of wireless networks, the importance of radiated power has made power control an essential component of network design. In [13], we analyzed the problem of power control in large, random wireless networks that are obtained by “erasing” a finite fraction of nodes from a regular $d$-dimensional lattice of $N$ transmit-receive pairs. Drawing on tools and ideas from statistical physics, we showed that this problem can be mapped to the Anderson impurity model for diffusion in random media; in this way, by employing the so-called coherent potential approximation (CPA) method, we calculated the average power in the system (and its variance) for 1-D and 2-D networks. In this regard, even though infinitely large systems are always unstable beyond a critical value of the users' SINR target, finite systems remain stable with high probability even beyond this critical SINR threshold. We calculated this probability by analyzing the density of low lying eigenvalues of an associated random Schrödinger operator, and we showed that the network can exceed this critical SINR threshold by a factor of at least $O\left({(logN)}^{-2/d}\right)$ before undergoing a phase transition to the unstable regime.