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.