  Consider a wireless network of transmitter-receiver pairs where the
transmitters adjust their powers to maintain a target SINR level in the
presence of interference. In this paper, we analyze the optimal power vector
that achieves this target in large, random networks obtained by "erasing" a
finite fraction of nodes from a regular lattice of transmitter-receiver pairs.
We show that this problem is equivalent to the so-called Anderson model of
electron motion in dirty metals which has been used extensively in the analysis
of diffusion in random environments. A standard approximation to this model is
the so-called coherent potential approximation (CPA) method which we apply to
evaluate the first and second order intra-sample statistics of the optimal
power vector in one- and two-dimensional systems. This approach is equivalent
to traditional techniques from random matrix theory and free probability, but
while generally accurate (and in agreement with numerical simulations), it
fails to fully describe the system: in particular, results obtained in this way
fail to predict when power control becomes infeasible. In this regard, we find
that the infinite system is always unstable beyond a certain value of the
target SINR, but any finite system only has a small probability of becoming
unstable. This instability probability is proportional to the tails of the
eigenvalue distribution of the system which are calculated to exponential
accuracy using methodologies developed within the Anderson model and its ties
with random walks in random media. Finally, using these techniques, we also
calculate the tails of the system's power distribution under power control and
the rate of convergence of the Foschini-Miljanic power control algorithm in the
presence of random erasures. Overall, in the paper we try to strike a balance
between intuitive arguments and formal proofs.
