  Reconfigurable intelligent surface (RIS) technology is emerging as a
promising technique for performance enhancement for next-generation wireless
networks. This paper investigates the physical layer security of an
RIS-assisted multiple-antenna communication system in the presence of random
spatially distributed eavesdroppers. The RIS-to-ground channels are assumed to
experience Rician fading. Using stochastic geometry, exact distributions of the
received signal-to-noise-ratios (SNRs) at the legitimate user and the
eavesdroppers located according to a Poisson point process (PPP) are derived,
and closed-form expressions for the secrecy outage probability (SOP) and the
ergodic secrecy capacity (ESC) are obtained to provide insightful guidelines
for system design. First, the secrecy diversity order is obtained as
$\frac{2}{\alpha_2}$, where $\alpha_2$ denotes the path loss exponent of the
RIS-to-ground links. Then, it is revealed that the secrecy performance is
mainly affected by the number of RIS reflecting elements, $N$, and the impact
of the number of transmit antennas and transmit power at the base station is
marginal. In addition, when the locations of the randomly located eavesdroppers
are unknown, deploying the RIS closer to the legitimate user rather than to the
base station is shown to be more efficient. Moreover, it is also found that the
density of randomly located eavesdroppers, $\lambda_e$, has an additive effect
on the asymptotic ESC performance given by
$\log_2{\left({1}/{\lambda_e}\right)}$. Finally, numerical simulations are
conducted to verify the accuracy of these theoretical observations.
