  Reconfigurable Intelligent Surfaces (RISs), comprising large numbers of
low-cost and almost passive metamaterials with tunable reflection properties,
have been recently proposed as an enabling technology for programmable wireless
propagation environments. In this paper, we present asymptotic closed-form
expressions for the mean and variance of the mutual information metric for a
multi-antenna transmitter-receiver pair in the presence of multiple RISs, using
methods from statistical physics. While nominally valid in the large system
limit, we show that the derived Gaussian approximation for the mutual
information can be quite accurate, even for modest-sized antenna arrays and
metasurfaces. The above results are particularly useful when fast-fading
conditions are present, which renders instantaneous channel estimation
extremely challenging. We find that, when the channel close to an RIS is
correlated, for instance due to small angle spread, which is reasonable for
wireless systems with increasing carrier frequencies, the communication link
benefits significantly from statistical RIS phase optimization, resulting in
gains that are surprisingly higher than the nearly uncorrelated case. Using our
novel asymptotic properties of the correlation matrices of the impinging and
outgoing signals at the RISs, we can optimize the metasurfaces without
brute-force numerical optimization. Furthermore, when the desired reflection
from any of the RISs departs significantly from geometrical optics, the
metasurfaces can be optimized to provide robust communication links, without
significant need for their optimal placement.
