The legendre transform of two replicas of the Sherrington-Kirkpatrick spin glass model: A free energy inequality

We prove a variational inequality linking the values of the free energy per site at different temperatures. This inequality is based on the Legendre transform of the free energy of two replicas of the system. We prove that equality holds when β ≤ 1/√2 and fails when 1/√2 < β ≤ 1. We deduce from this that the mean entropy per site of the uniform distribution with respect to the distribution of the coupling σ¹_iσ²_i = ψ_i between two replicas is null when 0 ≤ β ≤ 1/√2 and strictly positive when 1/√2 < β ≤ 1. We exhibit thus a new secondary critical phenomenon within the high temperature region 0 ≤ β ≤ 1. We give an interpretation of this phenomenon showing that the fluctuations of the law of the coupling with the interactions remains strong in the thermodynamic limit when β > 1/√2. We also use our inequality numerically within the low temperature region to improve (slightly) the best previously known lower bounds for the free energy and the ground state energy per site.