  In this paper, we study physical layer security for the downlink of cellular
networks, where the confidential messages transmitted to each mobile user can
be eavesdropped by both (i) the other users in the same cell and (ii) the users
in the other cells. The locations of base stations and mobile users are modeled
as two independent two-dimensional Poisson point processes. Using the proposed
model, we analyze the secrecy rates achievable by regularized channel inversion
(RCI) precoding by performing a large-system analysis that combines tools from
stochastic geometry and random matrix theory. We obtain approximations for the
probability of secrecy outage and the mean secrecy rate, and characterize
regimes where RCI precoding achieves a nonzero secrecy rate. We find that
unlike isolated cells, the secrecy rate in a cellular network does not grow
monotonically with the transmit power, and the network tends to be in secrecy
outage if the transmit power grows unbounded. Furthermore, we show that there
is an optimal value for the base station deployment density that maximizes the
secrecy rate, and this value is a decreasing function of the signal-to-noise
ratio.
