THE LEGENDRE TRANSFORM OF 2 REPLICAS OF THE SHERRINGTON-KIRKPATRICK SPIN-GLASS MODEL - A FREE-ENERGY INEQUALITY

We prove a variational inequality linking the values of the free energ y per site at different temperatures, This inequality is based on the Legendre transform of the free energy of two replicas of the system, W e prove that equality holds when beta less than or equal to 1/root 2 a nd fails when 1/root 2 < beta less than or equal to 1. We deduce from this that the mean entropy per site of the uniform distribution with r espect to the distribution of the coupling sigma(i)(1) sigma(i)(2) = p si(i) between two replicas is null when 0 less than or equal to beta l ess than or equal to 1/root 2 and strictly positive when 1/root 2 < be ta less than or equal to 1. We exhibit thus a new secondary critical p henomenon within the high temperature region 0 less than or equal to b eta less than or equal to 1. We give an interpretation of this phenome non showing that the fluctuations of the :Law of the coupling with the interactions remains strong in the thermodynamic limit when beta > 1/ root 2. we also use our inequality numerically within the low temperat ure region to improve (slightly) the best previously :known lower boun ds for the free energy and the ground state energy per site.