On Guerra's broken replica-symmetry bound

Consider a random Hamiltonian H-N ((σ) over right arrow) for (σ) over right arrow is an element of Sigma(N) = {0,1}(N). We assume that the family (HN (a)) is jointly Gaussian centered and that for (σ) over right arrow (1), (σ) over right arrow (2) is an element of Sigma(N), N-1 E H-N ((σ) over right arrow (1)) H-N ((σ) over right arrow (2)) = xi(N-1 Sigma(i)less than or equal to(N)sigma(i)(1)sigma(i)(2)) for a certain function on R. F. Guerra proved the remarkable fact that the free energy of the system with Hamiltonian H-N ((σ) over right arrow (1)) + h Sigma(i)less than or equal to(N) sigma(i) is bounded below by the free energy of the Parisi solution provided that xi is convex on R. We prove that this fact remains (asymptotically) true when the function is only assumed to be convex on R+. This covers in particular the case of the p-spin interaction model for any p. (C) 2003 Academie des sciences. Published by Editions scientificlues et medicales Elsevier SAS. All rights reserved.