The high temperature region of the Viana-Bray diluted spin glass model

In this paper, we study the high temperature or low connectivity phase of the Viana - Bray model in the absence of magnetic field. This is a diluted version of the well known Sherrington - Kirkpatrick mean field spin glass. In the whole replica symmetric region, we obtain a complete control of the system, proving annealing for the infinite volume free energy and a central limit theorem for the suitably rescaled fluctuations of the multi-overlaps. Moreover, we show that free energy fluctuations, on the scale 1/N, converge in the infinite volume limit to a non-Gaussian random variable, whose variance diverges at the boundary of the replica-symmetric region. The connection with the fully connected Sherrington Kirkpatrick model is discussed.