Fluctuations and thermodynamic variables in mean field spin glass models

Abstract: We present two rigorous results on the Sherrington-Kirkpatrick mean field model for spin glasses, proven by elementary methods, based on properties of fluctuations, with respect to the external quenched noise, of the thermodynamic variables and order parameters. The first result gives the uniform convergence of the quenched average of the free energy in the thermodynamic limit to its annealed approximation, in the high temperature regime, including the assumed critical point beta=1 in our notations. The second result shows that the free energy can be expressed through a functional order parameter, of the type introduced by Parisi in the frame of the replica symmetry breaking method. The functional order parameter is implicitely given in terms of fluctuations of thermodynamic variables.