  This work focuses on the downlink of a single-cell multi-user system in which
a base station (BS) equipped with $M$ antennas communicates with $K$
single-antenna users through a reconfigurable intelligent surface (RIS)
installed in the line-of-sight (LoS) of the BS. RIS is envisioned to offer
unprecedented spectral efficiency gains by utilizing $N$ passive reflecting
elements that induce phase shifts on the impinging electromagnetic waves to
smartly reconfigure the signal propagation environment. We study the minimum
signal-to-interference-plus-noise ratio (SINR) achieved by the optimal linear
precoder (OLP), that maximizes the minimum SINR subject to a given power
constraint for any given RIS phase matrix, for the cases where the LoS channel
matrix between the BS and the RIS is of rank-one and of full-rank. In the
former scenario, the minimum SINR achieved by the RIS-assisted link is bounded
by a quantity that goes to zero with $K$. For the high-rank scenario, we
develop accurate deterministic approximations for the parameters of the
asymptotically OLP, which are then utilized to optimize the RIS phase matrix.
Simulation results show that RISs can outperform half-duplex relays with a
small number of passive reflecting elements while large RISs are needed to
outperform full-duplex relays.
