Upper and Lower Bounds to the Free Energy Density of a One-dimensional Spin Glass Model

In this paper, a one-dimensional short-ranged spin-glass model in a random magnetic field is studied. Noticing convexity of the partition function, we use Jensen's inequality to obtain a lower bound to the averaged free energy density and the slope inequality for an upper bound. These bounds are derived under very general conditions and they can be used as tests for various approximate methods such as the replica method.