Pierluigi Contucci, Sandro Graffi Convex Replica Simmetry Breaking from Positivity and Thermodynamic Limit (16K, latex) ABSTRACT. Consider a correlated Gaussian random energy model built by successively adding one particle (spin) into the system and imposing the positivity of the associated covariance matrix. We show that the validity of a recently isolated condition ensuring the existence of the thermodynamic limit forces the covariance matrix to exhibit the Parisi replica symmetry breaking scheme with a convexity condition on the matrix elements.