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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.