Self-averaging in the statistical mechanics of some lattice models

We discuss self-averaging of thermodynamic properties in some random lattice models. In particular, we investigate when self-averaging (in an almost sure sense) of the free energy implies self-averaging of the energy and heat capacity, and we discuss the connection between self-averaging in the almost sure sense, and self-averaging in an L p sense. Under quite general conditions we show that the average of the finite size heat capacity converges to the second derivative of the limiting quenched average free energy. We consider the application of these ideas to the problem of adsorption of a random copolymer at a surface, and to some related systems.