The Legendre transform of two replicas of the Sherrington-Kirkpatrick spin glass model

Abstract

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 whenβ≤1/\(\sqrt 2 \) and fails when 1/\(\sqrt 2 \) 1/\(\sqrt 2 \). 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.